Abstract

In this article, we study the existence of normalized solutions to the following mixed nonlinear Choquard equation with exponential growth \[\begin{align} \left\{ \begin{aligned} \mathcal{L}u+\lambda u \; &=\; \Lambda(I_{\alpha}\ast F(u))F'(u) \text{ in }\mathbb{R}^{2},\\ \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align}\] where \(\mathcal{L}= -\Delta+(-\Delta)^s\), \(0<s<1\), \(a>0\), \(I_{\alpha}\) is the Riesz potential of order \(\alpha\in (0,2)\), \(\Lambda>0\) is a parameter and \(\lambda\in \mathbb{R}\) appears as a Lagrange multiplier. Here, the nonlinearity \(F\) has exponential growth in \(\mathbb{R}^{2}\). Using variational methods, we prove the existence of normalized solution in the Pohožaev manifold. Moreover, we discuss the regularity result and the construction of the Pohožaev identity, essential for the existence.

1 Introduction↩︎

The aim of this article is to analyze a mixed nonlinear Choquard equation with exponential growth, focusing on the existence of normalized solutions \[\label{aa} \left\{ \begin{align} \mathcal{L}u+\lambda u \; &=\; \Lambda(I_{\alpha}\ast F(u))F'(u) \text{ in }\mathbb{R}^{2},\\ \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{align} \right.}\tag{1}\] where \(a>0\) is a constant, \(\Lambda>0\) is a parameter, \(I_{\alpha}\) is the Riesz potential of order \(\alpha\in (0,2)\) defined as: \[\begin{align} \begin{aligned} I_{\alpha}(x)=\frac{\Gamma(\frac{2-\alpha}{2})}{\Gamma(\frac{\alpha}{2})2^{\alpha}\pi|x|^{2-\alpha}}:=\frac{A_{\alpha}}{|x|^{2-\alpha}}, \quad x\in \mathbb{R}^{2}\backslash\{0\}, \end{aligned} \end{align}\] the mixed diffusion type operator \(\mathcal{L}=-\Delta+(-\Delta)^s\) with \[(-\Delta)^su(x)=C(2,s)\text{P.V.}\int_{\mathbb{R}^2}\frac{u(x)-u(y)}{|x-y|^{2+2s}}dy, \text{ for } s\in (0,1),\] here \(C(2, s)\) is the normalizing constant given by \[C(2,s)=\left(\int_{\mathbb{R}^2}\frac{1-\cos(x)}{|x|^{2+2s}}dx\right)^{-1},\] and P.V. is the abbreviation for principal value. For the sake of our convenience, we consider \(C(2,s)=1\). Denoting \(f=F'\), we assume that \(f\) is a continuous non decreasing function that satisfies the following conditions:

\(\displaystyle \lim_{t \to 0} \frac{f(t)}{|t|^\tau} = 0\), for some, \(\tau > 3;\)
there exists \(\theta>2+\frac{\alpha}{2}>1\) such that \(f(t)t\geq \theta F(t)> 0\), for all \(t\neq 0\);
there exist constants \(\tilde{\sigma}>2+\frac{\alpha}{2}\) and \(\mu>0\) such that \[F(t)\geq \mu\, |t|^{\tilde{\sigma}}, \quad \text{for all } t \in \mathbb{R};\]
define \(\tilde{F}(t)=f(t)t-\left(\frac{2+\alpha}{2}\right)F(t)\) for \(t\in\mathbb{R}\). Then \(\frac{\tilde{F}(t)}{t^{2+\frac{\alpha}{2}}}\) is nondecreasing on \((0,\infty)\) and nonincreasing on \((-\infty, 0)\).

In the subcritical case, we further assume that \(f\) satisfies:

\(f\) has exponential subcritical growth, that is, for every \(\gamma>0\), \[\begin{align} \begin{aligned} \lim_{|t|\rightarrow +\infty}\frac{|f(t)|}{e^{\gamma|t|^{2}}}=0. \end{aligned} \end{align}\]

We also consider the case of exponential critical growth in \(\mathbb{R}^{2}\). It is well known that in dimension two, the natural growth condition is governed by the Trudinger-Moser inequality. In the critical case, we assume that

\(f\) has \(\gamma_{0}\) exponential critical growth at \(+\infty\) in the sense of the Trudinger-Moser inequality, that is, there exists \(\gamma_{0}>0\) such that \[\begin{align} \lim_{|t|\to +\infty}\frac{|f(t)|}{e^{\gamma |t|^{2}}}= \begin{cases} 0, & \text{for all } \gamma > \gamma_{0},\\ +\infty, & \text{for all } \gamma < \gamma_{0}. \end{cases} \end{align}\]

Remark 1. [1] From [f1]-[f2], if \(f\) has subcritical exponential growth at \(+\infty\), then for fixed \(q>2;\tau>3\), and for any \(\varepsilon>0;\gamma>0\), there exists a constant \(\kappa_{\varepsilon}= \kappa_{\varepsilon} (q,\gamma,\varepsilon)>0\), such that \[\label{snonb} | f(t)|\leq\varepsilon|t|^{\tau}+\kappa_{\varepsilon}|t|^{q-1}\bigl(e^{\gamma |t|^{2}}-1\bigr), \quad \forall\, t\in \mathbb{R},\qquad{(1)}\] and \[\label{snona} |F(t)|\leq\varepsilon|t|^{\tau+1}+\kappa_{\varepsilon}|t|^{q}\bigl(e^{\gamma |t|^{2}}-1\bigr), \quad \forall\, t\in \mathbb{R}.\qquad{(2)}\] Similarly, if \(f\) has critical exponential growth at \(+\infty\) with critical exponent \(\gamma_{0}\), then for fixed \(q>2;\tau>3\), and for any \(\varepsilon>0;\gamma>\gamma_{0}\), there exists a constant \(\kappa_{\varepsilon}= \kappa_{\varepsilon}(q,\gamma,\varepsilon)>0\), such that \[\label{nonb} | f(t)|\leq\varepsilon|t|^{\tau}+\kappa_{\varepsilon}|t|^{q-1}\bigl(e^{\gamma |t|^{2}}-1\bigr), \quad \forall\, t\in \mathbb{R},\qquad{(3)}\] and \[\label{nona} |F(t)|\leq\varepsilon|t|^{\tau+1}+\kappa_{\varepsilon}|t|^{q}\bigl(e^{\gamma |t|^{2}}-1\bigr), \quad \forall\, t\in \mathbb{R}.\qquad{(4)}\]

Throughout this paper, we assume that if the nonlinearity \(f\) satisfies either [f5] or [f6], that is, \(f\) has exponential subcritical or critical growth. This distinction is closely related to the Trudinger-Moser inequality; see [2], [3].

Equations involving nonlinearity of the form \((I_{\alpha}*F(u))F'(u)\) are called Choquard equation, as in 1976, Choquard, at the Symposium on Coulomb Systems utilised the energy functional associated to equation \[\left\{ \begin{array}{rl} & -\Delta u + u = (I_2*|u|^2)u\;\;\text{in } \mathbb{R}^3,\\ & u \in H^1(\mathbb{R}^3), \end{array} \right.\] to examine a viable approximation to Hartree-Fock theory for a one-component plasma (see [4]). The equation has various other applications in quantum physics, for instance, it is used to characterise an electron confined within its own vacancy, see [5] and related sources. Several works have ever since conducted research on the existence, multiplicity, and qualitative characteristics of the solution to the Choquard-type equations as detailed in [6]–[8].

The study of mixed operators of the type \(\mathcal{L}\) as in the problem 1 is motivated by several applications where such a kind of operators are naturally generated, including the theory of optimal searching, biomathematics, and animal foraging, for which we refer to [9]–[11]. In applied sciences, they are used for investigating the changes in physical phenomena that have both local and nonlocal effects. For instance, they are present in bi-modal power law distribution systems, see [12]. Furthermore, they are present in models that are derived from the combination of two distinct scaled stochastic processes. The aim of our work is to find the standing wave solution to Nonlinear Schr\(\ddot{\text{o}}\)dinger Equation. A standing wave solution for a nonlinear Schr\(\ddot{\text{o}}\)dinger (NLS) equation driven by mixed local and nonlocal operators is given as follows: \[\label{NLS95eq} i\frac{\partial \psi}{\partial t}=-\Delta \psi +(-\Delta)^s\psi -\mu (I_{\alpha}*F(\psi))f(\psi),\tag{2}\] is of the form \(\psi(x,t)=e^{-i\lambda t}u(x)\), where \(\lambda\in \mathbb{R}\) and \(u\in H^1(\mathbb{R}^N)\) solves: \[\label{1462} -\Delta u +(-\Delta)^su=\lambda u +\mu (I_{\alpha}*F(u))f(u) \text{ in }\mathbb{R}^N.\tag{3}\] While addressing solutions to 3 , there exist two schools of thought. The initial approach involves fixing a \(\lambda \in \mathbb{R}\) and thereafter looking for the critical points of the associated functional; this method has already been extensively employed, see for instance [7], [13], [14] and references therein. The other method is to fix the \(L^2\)-norm and look for the solution to the following constrained problem: \[\left\{ \begin{array}{rcl} -\Delta u+(-\Delta)^su & = & \lambda u +\mu(I_{\alpha}*F(u))F'(u) \text{ in } \mathbb{R}^N,\\ | u |_2^2 & = & \tau^2, \end{array} \right.\] called the normalized solution or the solution with prescribed mass. The physical motivation to study this problem arises from the fact that its solution provides stationary states of a nonlinear Schr\(\ddot{\text{o}}\)dinger equation with a prescribed \(L^2\)-norm, which represents an entity that satisfies dynamic conditions and has a fixed mass. The study of normalized solutions can be dated back to the pioneering work in [15], where Jeanjean obtained the existence of radial solutions for \[\label{Norm95sol} \left\{ \begin{array}{rl} -\Delta u & = \lambda u +g(u)\;\;\text{in } \mathbb{R}^N,\\ | u |_2^2 & = c, \end{array} \right.\tag{4}\] under some assumptions on \(g\). In [16], the existence of infinite solutions to 4 under same assumptions has been shown. Further in [17], the normalized solutions are discussed and described in the case of bounded domains with Dirichlet boundary conditions. Considering the domain to be the unit ball and \(g(x)=|x|^{p-1}x\), the existence of normalized solutions has been seen for \(p\) lying in \((1,1+\frac{4}{N}),\; (1+\frac{4}{N}, 2^*-1)\) and \(p=1+\frac{4}{N}\) under some conditions on \(c\). Moreover, the problem in general bounded domains has been dealt by the authors in [18]. Recently such problems involving fixed mass constraints have attracted many researchers; see, for instance, the work of [19]–[24], where authors studied the existence, multiplicity and regularity of normalized solutions for some nonlinear Schr\(\ddot{\text{o}}\)dinger equations with several local and nonlocal nonlinearities. Normalized solutions are also taken into consideration in the study of quadratic ergodic mean field games systems, see in particular [25].
Almost all of the above-discussed studies deal in higher dimensions, that is, for \(N>2\). For \(N=2\), the critical exponent turns out to be \(\infty\), which complicates the problem further and, at the same time, makes it more intriguing for researchers to work on, as it suggests that traditional methods of analysis may break down and require new approaches to comprehend the behaviour of the system. Recently, some authors have worked in order to tackle this issue; for instance, see the work of Deng and Yu in [26], where they studied the problem: \[\left\{ \begin{array}{rl} -\Delta u +\lambda u & =(I_{\alpha}*F(u))f(u) \text{ in } \mathbb{R}^2;\\ |u|_2 &= a, \end{array} \right.\] and proved the existence of the normalized solution as well as the ground state solution. Further, Shen and Squassina in [27] discussed the existence and concentrating behaviour of the normalized solution to a quite general problem. Moreover, the normalized solutions for some Kirchhoff-Choquard equations have also been studied; for instance, see the works in [28], [29] and references therein. Motivated by the above well-established literature, we would like to study the existence and regularity of normalized solutions to an exponential growth The Choquard equation driven by a mixed local-nonlocal operator in \(\mathbb{R}^2\).

Considering the solution space to be \(H^1(\mathbb{R}^2)\) and the functional settings as discussed in Section 2, we begin with highlighting our main results:

Remark 2. We note that functions satisfying [f1]-[f4] and [f5] are available in the literature; see [1].

Under the above assumptions [f1]-[f6], our first and second results are the following.

Theorem 1. Suppose that \(f\) satisfies [f1]-[f2] and [f5]. Then, for every \(a>0\), the problem 1 admits a normalized weak solution \((u,\lambda)\in H^1(\mathbb{R}^2)\times \mathbb{R}\) with \(\lambda>0\) for sufficiently large \(\Lambda\). Moreover, if [f4] holds, then this solution can be chosen as a normalized ground state solution of 1 .

Theorem 2. Assume that \(f\) satisfies [f1]-[f3] and [f6]. If \[a^{2} <\frac{(2+\alpha)\pi}{\gamma_{0}},\] then there exists \(\mu^*=\mu^*(a)>0\) such that, for every \(\mu\geq \mu^*\), the problem 1 admits a normalized weak solution \((u, \lambda)\in H^1(\mathbb{R}^2)\times \mathbb{R}^+\). Moreover, if [f4] also holds, then this solution can be chosen as a normalized ground state solution of 1 .

Main Novelty and Strategy of the Proofs: Before going into the proofs, we briefly explain the main idea of our approach. The main novelty of the present work lies in the study of normalized solutions for a mixed local-nonlocal Choquard equation with exponential growth in \(\mathbb{R}^2\). To the best of our knowledge, this is the first result combining a mixed operator of the form \(\mathcal{L}= -\Delta + (-\Delta)^s\), a nonlocal Choquard-type nonlinearity, and exponential growth in the sense of the Trudinger-Moser inequality under a mass constraint. This setting introduces significant analytical difficulties due to the simultaneous presence of exponential nonlinearity, nonlocal convolution terms, and the lack of compactness in \(\mathbb{R}^2\).

The problem has a variational structure, so normalized solutions of 1 can be obtained as critical points of the functional \(J\) defined in 5 , restricted to the \(L^2\)-sphere \(\mathcal{S}(a)\). Since the equation is autonomous, we work in the radial space \(H^1_{rad}(\mathbb{R}^2)\), where the embedding into \(L^q(\mathbb{R}^2)\) is compact for all \(q\in(2,\infty)\). By Palais’ principle of symmetric criticality (see [30]), solutions found in the radial space are also solutions in the whole space \(H^1(\mathbb{R}^2)\). Under assumptions [f1] and [f6], the functional \(J\) is well-defined and of class \(C^1\), thanks to Proposition 3 and the Hardy-Littlewood-Sobolev inequality (see Proposition 4). Moreover, the nature of the problem gives rise to a mountain pass geometry (Lemma 4). Following [15], this ensures the existence of a Palais-Smale \((PS)\) sequence at the level \(\gamma(a)\).

A key novelty in our approach is the adaptation of the Pohožaev manifold technique to the mixed local-nonlocal framework with exponential growth, which requires delicate estimates compatible with the Trudinger-Moser setting. A key step to ensure existence is to obtain an upper bound for the mountain pass level. This estimate allows us to prove the strong convergence of the \((\mathrm{PS})\)-sequence to a nontrivial critical point of \(J\) on \(\mathcal{S}(a)\) (see Section 3). Finally, to show that this solution is a ground state, we use the monotonicity assumption [f4] and prove that along suitable fiber paths the functional attains its maximum at a unique point, see (Lemma 5), belonging to the Pohožaev manifold \(\{\mathcal{P}(a)\}\). Once this structure is established, we compare the mountain pass level \(\gamma(a)\) with the least energy level \(m(a)\), showing that they coincide and thus characterize the ground state energy.

The paper is organized as follows. In Section 2, we present the functional framework, the Trudinger-Moser type estimates, and the Pohožaev identity associated with the problem. Section 3 is devoted to the minimax construction and the compactness analysis of the corresponding Palais-Smale \((PS)\) sequence. More precisely, in Subsection 3.1 we establish the mountain-pass geometry and define the minimax level \(\gamma(a)\), while in Subsection 3.2 we study the compactness properties of the Palais-Smale sequence at this level. Section 4 contains the proofs of the main existence results. The proof of Theorem 1 deals with the subcritical exponential case, whereas the proof of Theorem 2 treats the critical exponential case and includes the critical minimax estimates. Finally, Section 5 is devoted to the regularity result and to the derivation of the Pohožaev identity.

Notation. Throughout this paper, unless otherwise stated, we use the following notation:

\(\|\cdot\|\) denotes the norm for the Sobolev space \(H^1(\mathbb{R}^2)\);
\(|\cdot|_p\) denotes the usual norm in the Lebesgue space \(L^{p}(\mathbb{R}^2)\), for \(p \in [1,+\infty]\);
\(|u|^a\) denotes the pointwise power \(|u|^a=|u(x)|^a\), for \(1\leq a<+\infty\);
\(o_n(1)\) denotes a sequence such that \(o_n(1)\to 0\) as \(n \to +\infty\);
\(C, C_1, C_2, \dots\) denote positive constants.

2 Functional Framework↩︎

In this section, we introduce the variational framework for problem 1 and discuss some results that will be useful throughout the paper. We start with the following embedding result.

Lemma 1. [31] Let \(0<s<1\). Then \(H^1(\mathbb{R}^2)\) is continuously embedded into \(H^s(\mathbb{R}^2)\).

The presence of both local and nonlocal operators in 1 naturally leads us to consider the space \(H^1(\mathbb{R}^2)\) to be the solution space, with the inner product \[(u,v)= \int_{\mathbb{R}^2}\nabla u\cdot\nabla v\,dx + \ll u, v \gg +\int_{\mathbb{R}^2}uv\,dx,\] where \[\ll u,v\gg:=\frac{1}{2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{2+2s}}\,dx\,dy\] and the associated norm \[\|u\|=(u,u)^{1/2}=\left(|\nabla u|_2^2+\frac{[u]_s^2}{2}+|u|_2^2\right)^{\frac{1}{2}},\] with \[[u]_s^2=\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}}\,dx\,dy.\] Next, we recall the Trudinger-Moser inequality in \(\mathbb{R}^2\) that will help us to deal with the exponential growth of \(f\).

Proposition 3. [32], [33] If \(\gamma>0\) and \(u\in H^{1}(\mathbb{R}^{2})\), then \[\begin{align} \begin{aligned} \int_{\mathbb{R}^{2}}\Big(e^{\gamma |u|^{2}}-1\Big)dx<\infty. \end{aligned} \end{align}\] Moreover, if \(\gamma<4\pi\) and \(|u|_{2}\leq M<\infty\), then there exists a constant \(\mathcal{C}(M,\gamma)>0\) such that \[\begin{align} \begin{aligned} \sup_{|\nabla u|^{2}_{2}\leq1, |u|_{2}\leq M } \int_{\mathbb{R}^{2}}\Big(e^{\gamma |u|^{2}}-1\Big)dx <\mathcal{C}(M,\gamma). \end{aligned} \end{align}\]

We now introduce and prove a Trudinger-Moser type inequality for the mixed local-nonlocal setting.

Proposition 4. Let \(u\in H^{1}(\mathbb{R}^{2})\). Then the following assertions hold:

If \(\gamma<4\pi\) and \(\|u\|\leq 1\), then there exists a constant \(C_{\gamma}>0\), depending only on \(\gamma\), such that \[\int_{\mathbb{R}^{2}}\left(e^{\gamma |u|^{2}}-1\right)\,dx\leq C_{\gamma}.\]
More generally, if \(u\in H^1(\mathbb{R}^{2})\) satisfies \[\gamma\|u\|^{2}<4\pi,\] then there exists a constant \(C>0\) such that \[\int_{\mathbb{R}^{2}}\left(e^{\gamma |u|^{2}}-1\right)\,dx\leq C.\]

Proof. If \(\gamma<4\pi\) and \(\|u\|\le 1\), then \[|\nabla u|_2\le \|u\|\le 1, \qquad |u|_2\le \|u\|\le 1.\] Thus Proposition 3 gives \[\int_{\mathbb{R}^2}\left(e^{\gamma |u|^2}-1\right)\,dx\le C_\gamma,\] and hence (i) follows. Finally, assume that \(\gamma\|u\|^2<4\pi\). If \(u=0\), the result is trivial. Otherwise set \(v=u/\|u\|\). Then \(\|v\|=1\). Let \[\beta:=\gamma\|u\|^2<4\pi.\] Applying (i) to \(v\) with exponent \(\beta\), we obtain \[\int_{\mathbb{R}^2}\left(e^{\beta |v|^2}-1\right)\,dx\le C_\beta.\] Since \(\gamma |u|^2=\beta |v|^2\), we get \[\int_{\mathbb{R}^2}\left(e^{\gamma |u|^2}-1\right)\,dx = \int_{\mathbb{R}^2}\left(e^{\beta |v|^2}-1\right)\,dx \le C_\beta.\] This proves (ii). ◻

We also recall the Hardy-Littlewood-Sobolev inequality; see [34].

Proposition 5. Let \(t_1,t_2>1\), \(0<\alpha<2\), with \(\frac{1}{t_1}+\frac{2-\alpha}{2}+\frac{1}{t_2}=2\). If \(f\in L^{t_1}(\mathbb{R}^{2})\) and \(g\in L^{t_2}(\mathbb{R}^{2})\), then there exists a constant \(C(t_1,\alpha,t_2)>0\) such that \[\begin{align} \begin{aligned} \int_{\mathbb{R}^{2}} (I_{\alpha}\ast f)g\, dx \leq C(t_1,\alpha,t_2)|f|_{t_1}|g|_{t_2}. \end{aligned} \end{align}\]

As a consequence of Proposition 5, the term \[\int_{\mathbb{R}^{2}} (I_{\alpha}\ast F(u))F(u)\, dx\] is well defined provided that \(F(u)\in L^{t}(\mathbb{R}^{2})\) with \(t>1\) satisfying \[\frac{2}{t}+\frac{2-\alpha}{2}=2,\] which yields \[F(u)\in L^{\frac{4}{2+\alpha}}(\mathbb{R}^{2}).\] Finally, we introduce the notion of weak solutions.

Definition 1. A function \(u \in H^1(\mathbb{R}^2)\) is said to be a weak solution of 1 if \(|u|_2^2 = a^2\) and \[\int_{\mathbb{R}^2} \nabla u \cdot \nabla v\,dx +\ll u,v \gg +\lambda \int_{\mathbb{R}^2}uv\,dx = \Lambda\int_{\mathbb{R}^2} (I_\alpha * F(u))\,f(u)v\,dx,\] for all \(v \in H^1(\mathbb{R}^2)\).

Solutions of 1 , called the normalized solutions, correspond to critical points of the energy functional \[J:H^1(\mathbb{R}^{2}) \to \mathbb{R}\] defined by \[\label{J} \begin{align} J(u) &=\frac{|\nabla u|_2^2}{2} +\frac{[u]^2_s}{4} -\frac{\Lambda}{2}\int_{\mathbb{R}^{2}} (I_{\alpha}\ast F(u))F(u)\,dx, \end{align}\tag{5}\] constrained to the following \(L^2\)- sphere \[\label{tau} \mathcal{S}(a):=\left\{u\in H^{1}(\mathbb{R}^{2}) : |u|_2^2 = a^2\right\}.\tag{6}\] For \(a>0\), if \(u\) is a critical point of the constrained functional \(J|_{\mathcal{S}(a)}\), i.e., \(J'|_{\mathcal{S}(a)}(u)=0,\) then, since \(\mathcal{S}(a)\) is regular, there exists some \(\lambda\in\mathbb{R}\) such that \((u,\lambda)\) solves 1 . Here, the parameter \(\lambda\in\mathbb{R}\) will appear as a Lagrange multiplier depending on the solution \(u\in H^{1}(\mathbb{R}^2)\) and is not a priori given. To check that the functional \(J\) is well defined, it is enough to see if \((I_{\alpha}*F(u))F(u)\in L^1(\mathbb{R}^2)\). Since, by using Proposition 5, ?? and H\(\ddot{\text{o}}\)lder’s inequality, for any \(t,t'>1\) satisfying \(\frac{1}{t}+\frac{1}{t'}=1\), we have: \[\begin{align} \int_{\mathbb{R^2} }(I_{\alpha}*F(u))F(u) & \leq & C(\alpha) \left(\int_{\mathbb{R}^{2}}|F(u)|^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{4}}\left(\int_{\mathbb{R}^{2}}|F(u)|^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{4}}\\ & = & C(\alpha) \left(|F(u)|^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}}\\ & \leq & C(\alpha) \left(\int_{\mathbb{R}^{2}}\left(\varepsilon|u|^{\tau+1}+C_{\varepsilon}|u|^q\left(e^{\gamma|u|^2}-1\right)\right)^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}}\\ & \leq & C'(\alpha)\left(\left(\int_{\mathbb{R}^{2}}\varepsilon^{\frac{4}{2+\alpha}}|u|^{(\tau+1)\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}} \right.\\ &&+ \left.\left(\int_{\mathbb{R}^{2}}C_{\varepsilon}^{\frac{4}{2+\alpha}}|u|^{\frac{4q}{2+\alpha}}\left(e^{\gamma|u|^2}-1\right)^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}}\right)\\ & \leq & C'(\alpha) \left(\varepsilon^2 |u|_{(\tau+1)\frac{4}{2+\alpha}}^{2(\tau+1)}+C_{\varepsilon}^2\left(\int_{\mathbb{R}^{2}}|u|^{\frac{4qt'}{2+\alpha}}\right)^{\frac{2+\alpha}{2t'}}\left(\int_{\mathbb{R}^{2}}\left(e^{\gamma|u|^2-1}\right)^{\frac{4t}{2+\alpha}}\right)^{\frac{2+\alpha}{2t}}\right)\\ & < & +\infty, \end{align}\] by the continuous inclusion of \(H^1(\mathbb{R}^2)\hookrightarrow L^r(\mathbb{R}^2)\) for all \(r\in [2,\infty)\), the fact that \[(e^s-1)^m \le C_m (e^{ms}-1), \qquad ~ m>1,~s\ge 0,\] and using the Proposition 3. Therefore, \(J\) is well defined and of class \(C^{1}\). However, it is well known that \(J\) does not satisfy the Palais-Smale condition. To deal with this lack of compactness, we exploit the associated Pohožaev identity, see Appendix for details, and study the functional on the Pohožaev Manifold \(\mathcal{P}(a)\).

Lemma 2. If \((u,\lambda)\) is any couple weakly solving problem 1 , then \(u \in \mathcal{P}(a)\), where \[\label{Pohozaev-manifold} \mathcal{P}(a) := \left\{ u\in \mathcal{S}(a)\;:\; P(u)=0 \right\},\qquad{(5)}\] where \[\label{onupqfvj} \begin{align} P(u) &:= |\nabla u|_{2}^{2} +\frac{s}{2}[u]_{s}^{2} +\Lambda\left(\frac{2+\alpha}{2}\right) \int_{\mathbb{R}^{2}}(I_{\alpha}*F(u))F(u)\,dx - \Lambda\int_{\mathbb{R}^{2}}(I_{\alpha}*F(u))f(u)u\,dx . \end{align}\qquad{(6)}\]

Proof. Indeed, testing the equation with the solution itself, one gets \[\label{eq952465} | \nabla u |_2^2 + \frac{1}{2}[u]_s^2 +\lambda | u |_2^2=\Lambda\int_{\mathbb{R}^N}(I_{\alpha}*F(u))f(u)udx.\tag{7}\] Also, as proved in Theorem 6 any solution to 1 must satisfy the following \[\label{eq952466} \left(\frac{1-s}{2}\right)[u]_s^2+\lambda \left| u \right|_2^2= \Lambda\left(\frac{2+\alpha}{2}\right)\int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u)dx.\tag{8}\] Using 7 in 8 we get \[| \nabla u |_2^2 +\frac{s}{2}[u]_s^2+ \Lambda\left(\frac{2+\alpha}{2}\right)\int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u)-\Lambda\int_{\mathbb{R}^2}(I_{\alpha}*F(u))f(u)u=0.\] , \(P(u) = 0\). ◻

Thus any solution of 1 lies on the Pohožaev manifold \(\mathcal{P}(a)\) and we define the ground state solution as follows:

Definition 2. Suppose, \(u\in H^1(\mathbb{R}^2)\) solves 1 , we call it normalized ground state if \(J(u)\) possesses the least energy among all normalized solutions, i.e., \[J(u) = \min\left\{ J(v): v\in \mathcal{P}(a),\;J'|_{\mathcal{P}(a)}(v)=0 \right\},\] in particular, if \(u\) satisfies \[J(u)=m(a):=\inf_{v\in\mathcal{P}(a)} J(v).\]

To overcome the lack of compactness due to whole space \(\mathbb{R}^2\), we restrict our analysis to the radial subspace \(H^{1}_{rad}(\mathbb{R}^{2})\) of \(H^{1}(\mathbb{R}^{2})\). Accordingly, we define \[\mathcal{S}_{r}(a):=\mathcal{S}(a)\cap H^{1}_{rad}(\mathbb{R}^{2}),\] where \[H^{1}_{rad}(\mathbb{R}^{2}) := \left\{ u \in H^{1}(\mathbb{R}^{2}) : u(x) = u(|x|),\;x \in \mathbb{R}^2 \right\}.\]

In the sequel, we shall use the radial constraint \(S_r(a)\) in order to recover compactness. The minimax construction and the Palais-Smale analysis will be carried out under a unified Trudinger-Moser threshold condition. By [f5], for \(\gamma>0\) sufficiently small, one has \(\gamma \| u\|^{2} < (2+\alpha)\pi,\) and \(u\) bounded in \(H^{1}(\mathbb{R}^{2})\). Similarly, by [f6], for \(\gamma>\gamma_{0}\) sufficiently close to \(\gamma_{0}\), we have \[\gamma \| u\|^{2} < (2+\alpha)\pi, \qquad \|u\|^{2}<\frac{(2+\alpha)\pi}{\gamma_{0}}.\] Therefore, in what follows, we work under the unified condition \(\gamma \| u\|^{2} < (2+\alpha)\pi.\)

3 Preliminaries for existence results↩︎

In this section, we develop the variational structure needed to obtain normalized solutions to the problem \((P_\lambda)\). This section is divided into two parts. In Subsection 3.1, we establish the minimax structure and define the mountain-pass level \(\gamma(a)\). In Subsection 3.2, we analyse the Palais-Smale sequence obtained at this level and prove the compactness properties required in the proofs of the main theorems.

3.1 The Minimax Approach↩︎

Defining the \(L^2\) norm preserving scaling, called the fibre map \[H(u,\sigma)(x):=e^\sigma u(e^\sigma x), \text{ for every } (u,\sigma)\in H^1(\mathbb{R}^2)\times\mathbb{R}.\] Clearly, \[|\nabla H(u,\sigma)|_2^2=e^{2\sigma}|\nabla u|_2^2;\;\;[H(u,\sigma)]_s^2=e^{2s\sigma}[u]_s^2,\;\;|H(u,\sigma)|_{\xi}^{\xi}=e^{(\xi-2)\sigma}|u|_{\xi}^{\xi} \text{ for any }\xi\geq2\] and \[\int_{\mathbb{R}^2}(I_{\alpha}*F(H(u,\sigma)))F(H(u,\sigma))dx=e^{-(2+\alpha)\sigma}\int_{\mathbb{R}^2}(I_{\alpha}*F(e^\sigma u))F(e^\sigma u)dx.\] We initiate by discussing the behaviour of the constrained functional \(J\) along this fibre map.

Lemma 3. Let \(u\in S_r(a)\) and [f1]-[f2] holds. Then,

as \(\sigma\rightarrow-\infty\), we have: \[\left\{ | \nabla H(u,\sigma)|_2^2 \rightarrow 0; \;[H(u,\sigma)]_s^2 \rightarrow 0 \text{ and } J(H(u,\sigma)) \rightarrow 0; \right\}\]
as \(\sigma\rightarrow +\infty\) we have: \[\left\{ | \nabla H(u,\sigma)|_2^2 \rightarrow +\infty; \;[H(u,\sigma)]_s^2 \rightarrow +\infty \text{ and } J(H(u,\sigma)) \rightarrow -\infty. \right\}\]

Proof. Clearly, \[|\nabla H(u,\sigma)|_2^2=e^{2\sigma}|\nabla u|_2^2 \rightarrow \left\{ \begin{array}{rl} +\infty & \text{ as } \sigma\rightarrow+\infty, \\ 0 & \text{ as } \sigma\rightarrow-\infty \end{array} \right.\] and

\[[ H(u,\sigma)]_s^2=e^{2s\sigma}[u]_s^2 \rightarrow \left\{ \begin{array}{rl} +\infty & \text{ as } \sigma\rightarrow+\infty, \\ 0 & \text{ as } \sigma\rightarrow-\infty. \end{array} \right.\] Next, we study the behaviour of \(J(H(u,\sigma))\). Precisely, we have: \[\label{J40H41} J(H(u,\sigma))= \frac{e^{2\sigma}}{2}|\nabla u|_2^2+\frac{e^{2s\sigma}}{4}[u]_s^2 -\frac{\Lambda}{e^{(2+\alpha)\sigma}}\int_{\mathbb{R}^2}(I_{\alpha}*F(e^{\sigma}u))F(e^{\sigma}u)dx.\tag{9}\] Now, using Proposition 5, ?? , ?? and H\(\ddot{\text{o}}\)lder’s inequality, we get: \[\begin{align} && \int_{\mathbb{R}^2}(I_{\alpha}*F(H(u,\sigma)))F(H(u,\sigma))\leq C_1\left(\int_{\mathbb{R}^2}|F(H(u,\sigma))|^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}}\\ &&\leq C_1\left(\int_{\mathbb{R}^2}|\varepsilon|H(u,\sigma)|^{\tau+1}+\kappa_{\varepsilon}|H(u,\sigma)|^q\left(e^{\gamma|H(u,\sigma)|^2}-1\right)|^{\frac{4}{2+\alpha}}dx\right)^{\frac{2+\alpha}{2}}\\ &&\leq C_2 \left(\varepsilon^{\frac{4}{2+\alpha}}\int_{\mathbb{R}^2}|H(u,\sigma)|^{\frac{(\tau+1)4}{2+\alpha}}+\kappa_{\varepsilon}^{\frac{4}{2+\alpha}}\int_{\mathbb{R}^2}|H(u,\sigma)|^{\frac{2q}{2+\alpha}}\left(e^{\gamma|H(u,\sigma)|^2}-1\right)^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}}\\ && \leq C_3 \varepsilon^2 |H(u,\sigma)|_{\frac{(\tau+1)4}{2+\alpha}}^{2(\tau+1)}+ C_3 \kappa_{\varepsilon}^2 \left(\left(\int_{\mathbb{R}^2}|H(u,\sigma)|^{\frac{4qt}{2+\alpha}}\right)^{\frac{1}{t}}\left(\int_{\mathbb{R}^2}\left(e^{\gamma |H(u,\sigma)|^2}-1\right)^{\frac{4t'}{2+\alpha}}\right)^{\frac{1}{t'}}\right)^{\frac{2+\alpha}{2}} \end{align}\] for some \(t,t'>1\) such that \(\frac{1}{t}+\frac{1}{t'}=1\). Choosing \(t'>1\) close to \(1\) such that \[\gamma t'\|H(u,\sigma)\|^{2}\leq (2+\alpha)\pi,\] which implies that \[\label{1Domina195mixed} \left( \int_{\mathbb{R}^{2}} \left(e^{\gamma|H(u,\sigma)|^2}-1\right)^{\frac{4t'}{2+\alpha}} \right)^{\frac{2+\alpha}{2t'}}\leq C_4\left(\int_{\mathbb{R}^2}\left(e^{\frac{4t'\gamma|}{2+\alpha}|H(u,\sigma)|^2}-1\right)dx\right)^{\frac{2+\alpha}{2t'}} \leq C.\tag{10}\] Thus, \[\begin{align} \int_{\mathbb{R}^2}(I_{\alpha}*F(H(u,\sigma)))F(H(u,\sigma)) & \leq & C_3 \varepsilon^2 |H(u,\sigma)|_{\frac{4(\tau+1)}{2+\alpha}}^{2(\tau+1)} +C_5 \kappa_{\varepsilon}^2 |H(u,\sigma)|_{\frac{4qt}{2+\alpha}}^{2q}\\ & = & C_3 \varepsilon^2 e^{(2\tau-\alpha)\sigma} |u|_{\frac{4(\tau+1)}{2+\alpha}}^{2(\tau+1)}+ C_5\kappa_{\varepsilon}^2 e^{\frac{(4qt-4-2\alpha)}{2t}\sigma}|u|_{\frac{4qt}{2+\alpha}}^{2q}\\ && \rightarrow 0 \text{ as } \sigma\rightarrow-\infty, \end{align}\] since \(2\tau-\alpha \text{ and }\frac{4qt-4-2\alpha}{2t}>0\) for \(\tau>3\) and \(q>2\). Therefore, \(J(H(u,\sigma))\rightarrow0\) as \(\sigma\rightarrow -\infty\). This proves \((i)\).

We now prove \((ii)\). Define \[g(z)=\int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(z))F(z)\,dx.\] Set \[w(t)=g\!\left(\frac{tu}{\|u\|}\right)=\int_{\mathbb{R}^2}\left(I_{\alpha}*F\left(\frac{tu}{\left\| u \right\|}\right)\right)F\left(\frac{tu}{\left\| u \right\|}\right)dx.\] By [f2], we know that \[\frac{w'(t)}{w(t)}\geq \frac{2\theta}{t} \qquad \text{for } t>0,\] which implies that \[g(tu)\geq g\!\left(\frac{u}{\|u\|}\right)\|u\|^{2\theta}t^{2\theta}.\] Therefore, taking \(t=e^{\sigma}\) by 9 we obtain \[\begin{align} J(H(u,\sigma)) &\leq C_{1}e^{2\sigma} + C_{2}e^{2s\sigma} - C_{3}e^{(2\theta-(2+\alpha))\sigma}. \end{align}\] Since \(0<s<1\) and \[2\theta-(2+\alpha)>2>2s\] the negative term dominates as \(\sigma\to +\infty\). Hence, \[J(H(u,\sigma))\to -\infty \qquad \text{as }\sigma\to +\infty.\] This proves \((ii)\) and completes the proof. ◻

Now, for any \(k>0\), we define: \[A_k:=\{u\in \mathcal{S}_r(a): |\nabla u|_2^2+[u]_s^2\leq k\}; \text{ and } B_k:=\{u\in \mathcal{S}_r(a): |\nabla u |^2_2+[u]_s^2 =4k\},\] and study the functional \(J\) constrained over these sets.

Lemma 4. There exists \(K(a)>0\) such that \(J(u)\), \(P(u)>0\) for all \(u\in A\) and \[0<\sup_{u\in A}J(u)\leq \inf_{u\in B} J(u),\] where \(A=A_{K(a)}\) and \(B=B_{K(a)}\).

Proof. Suppose \(u\in \mathcal{S}_r(a)\) be such that \(|\nabla u|_2^2+[u]_s^2=k\). Following the arguments of Lemma 3, we can find \(t,t'>1\) satisfying \(\frac{1}{t}+\frac{1}{t'}=1\) with \(t'\) close to \(1\) such that \[\int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u)\leq C_1\varepsilon^2 |u|_{\frac{4(\tau+1)}{2+\alpha}}^{2(\tau+1)}+C_2\kappa_{\varepsilon}^2 |u|_{\frac{4qt}{2+\alpha}}^{2q},\] moreover, by the following Gagliardo-Nirenberg inequality [35]: \[\label{1eq} |u|_{p}\leq C(p)|\nabla u|_2^{\theta}|u|_2^{1-\theta} \leq C(p)\left(|\nabla u|_2^2+[u]_s^2\right)^{\theta}|u|_2^{1-\theta}\text{ for all } p\geq 2, \text{ with } \theta=1-\frac{2}{p};\tag{11}\] we get \[\label{Lemma346295F} \int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u) \leq C_3 \varepsilon^2 a^{2+\alpha}k^{\frac{2\tau-\alpha}{2}}+C_4\kappa_{\varepsilon}^2a^{\frac{2+\alpha}{t}}k^{\frac{2qt-2-\alpha}{2t}}.\tag{12}\] Similarly, by ?? , ?? and following the arguments of Lemma 3 we get: \[\begin{align} \label{Lemma346295f40u41u} \int_{\mathbb{R}^2}(I_{\alpha}*F(u))f(u)u & \leq & C_5 \left(\int_{\mathbb{R}^2}|F(u)|^{\frac{4}{2+\alpha}} \right)^{\frac{2+\alpha}{4}}\left(\int_{\mathbb{R}^2}|f(u)u|^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{4}}\nonumber\\ & \leq & C_5\left(\int_{\mathbb{R}^2}|\varepsilon|u|^{\tau+1}+\kappa_{\varepsilon}^2|u|^q\left(e^{\gamma |u|^2}-1\right)|^{\frac{4}{2+\alpha}}\right)^{\frac{2+\alpha}{2}} \nonumber \\ & \leq & C_6\varepsilon^2 a^{2+\alpha}k^{\frac{2\tau-\alpha}{2}}+C_7\kappa_{\varepsilon}^2a^{\frac{2+\alpha}{t}}k^{\frac{2qt-2-\alpha}{2t}}. \end{align}\tag{13}\] By 12 and 13 we get: \[\begin{align} J(u) & = & \frac{|\nabla u|_2^2}{2}+\frac{[u]_s^2}{4}-\frac{\Lambda}{2}\int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u)dx\\ & \geq & \frac{k}{4}-\left(\frac{\Lambda}{2}C_3\varepsilon^2a^{2+\alpha}\right)k^{\frac{2\tau-\alpha}{2}}-\left(\frac{\Lambda}{2}C_4\kappa_{\varepsilon}^2a^{\frac{2+\alpha}{t}}\right)k^{\frac{2qt-2-\alpha}{2t}}, \end{align}\] and \[\begin{align} P(u) & = & |\nabla u|_2^2+\frac{s}{2}[u]_s^2 +\Lambda\left(\frac{2+\alpha}{2}\right)\int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u)dx -\Lambda \int_{\mathbb{R}^2}(I_{\alpha}*F(u))f(u)udx\\ & \geq & \frac{s}{2}k -\left({\Lambda}C_6\varepsilon^2a^{2+\alpha}\right)k^{\frac{2\tau-\alpha}{2}}-\left({\Lambda}C_7\kappa_{\varepsilon}^2a^{\frac{2+\alpha}{t}}\right)k^{\frac{2qt-2-\alpha}{2t}}. \end{align}\] Now, since \(t'\) is close to \(1\), clearly we can consider \(t>2\), which implies that \(\frac{2\tau-\alpha}{2}>1\) and \(\frac{2qt-2-\alpha}{2t}>1\). Hence we can find \(K(a)>0\) such that \[J(u) \text{ and } P(u)>0 \text{ for all } k\in [0,K(a)].\] Therefore, considering \(A=A_{K(a)}\), we get \(J(u)>0\) and \(P(u)>0\) for all \(u\in A\).

Since \(\tau>3\), \(q\) and \(t>2\), choosing \(K(a)\) sufficiently small, we obtain \[\frac{K(a)}{2}-\left(\frac{\Lambda}{2}C_3'\varepsilon^2a^{2+\alpha}\right)K(a)^{\frac{2\tau-\alpha}{2}}-\left(\frac{\Lambda}{2}C_4'\kappa_{\varepsilon}^2a^{\frac{2+\alpha}{t}}\right)K(a)^{\frac{2qt-2-\alpha}{2t}}>0\] which proves the desired result. ◻

As a direct consequence of Lemma 4, we have the following corollary.

Corollary 1. Assume that [f1]-[f2] hold. Let \(u\in \mathcal{S}_{r}(a)\). Then, \[J_*:= \inf \left\{ J(u): u\in\mathcal{S}_r(a),\; \int_{\mathbb{R}^2}|\nabla u|^2\,dx+[u]_s^2={ K(a)} \right\} >0.\]

Proof. Arguing as in the proof of Lemma 4, we obtain \[J(u)\geq \frac{K(a)}{4}-\left(\frac{\Lambda}{2}C_3\varepsilon^2a^{2+\alpha}\right)K(a)^{\frac{2\tau-\alpha}{2}}-\left(\frac{\Lambda}{2}C_4\kappa_{\varepsilon}^2a^{\frac{2+\alpha}{t}}\right)K(a)^{\frac{2qt-2-\alpha}{2t}}=\tilde{\delta}>0,\] for every \(u\in \mathcal{S}_r(a)\) such that \[|\nabla u|_2^2 +[u]_s^2=K(a).\] Hence, \[\begin{align} J_* & = & \inf\left\{ J(u): u\in\mathcal{S}_r(a),\; |\nabla u|_2^2+[u]_s^2={ K(a)} \right\}\\ & \geq & \tilde{\delta}>0. \end{align}\] This completes the proof. ◻

We are now in a position to define the minimax level. Following the ideas of Jeanjean [15], we define the class of admissible paths by \[\Gamma:= \left\{ h\in C([0,1],\mathcal{S}_{r}(a)): {h(0)\in A}\text{ and } {J(h(1))<0} \right\},\] where \(A\) is as defined in Lemma 4, and the corresponding mountain-pass level by \[\gamma(a):=\inf_{h\in\Gamma}\max_{t\in[0,1]}J(h(t)).\] We claim that \(\gamma(a)>0\). Indeed, let \(h\in\Gamma\) be arbitrary and define \[\phi_h(t):=|\nabla h(t)|_2^2+[h(t)]_s^2, \qquad t\in[0,1].\] Since \(h\in\Gamma\), we have \(h(0)\in A\), and hence \[\phi_h(0)=|\nabla h(0)|_2^2+[h(0)]_s^2\leq K(a).\] Also since \(J(h(1))<0\), we must have \(\phi_h(1)>K(a)\), otherwise we will contradict Lemma 4. Therefore, by continuity of \(h\) there exists \(t_0\in [0,1]\) such that \[|\nabla h(t_0)|_2^2+[h(t_0)]_s^2={K(a)}.\] Consequently, by Corollary 1 we get \[\max_{t\in[0,1]}J(h(t)) \geq J(h(t_0)) \geq J_*>0.\] Since \(h\in\Gamma\) was arbitrary, we conclude that \[\gamma(a) = \inf_{h\in\Gamma}\max_{t\in[0,1]}J(h(t)) \geq J_*>0.\] Moreover, we will see that \(\gamma(a)\) is equal to the infimum of the functional \(J\) over the Pohožaev manifold, for that, we first prove the following lemma.

Lemma 5. Assume that [f1]-[f2] and [f4] hold, and let \(u\in \mathcal{S}_{r}(a)\). Then the functional \[\widetilde{J}_{u}(\sigma):=J(H(u,\sigma)), \qquad \sigma\in \mathbb{R},\] admits a unique maximum point \(\sigma(u)\in \mathbb{R}\) such that \[H(u,\sigma(u))\in \mathcal{P}(a).\] Moreover, for \(v\in \mathcal{P}(a)\), \(\sigma(v)=0.\)

Proof. Let \(u\in S_r(a)\) and \(\sigma\in \mathbb{R}\), then we have: \[\begin{align} \tilde{J}(\sigma) & = & J(H(u,\sigma)) \\ & = & \frac{|\nabla H(u,\sigma)|_2^2}{2}+\frac{[H(u,\sigma)]_s^2}{4}-\frac{\Lambda}{2}\int_{\mathbb{R}^2}(I_{\alpha}*F(H(u,\sigma))F(H(u,\sigma))dx\\ & = & \frac{e^{2\sigma}}{2}|\nabla u |_2^2 +\frac{e^{2s\sigma}}{4}[u]_s^2 -\frac{\Lambda}{2e^{(2+\alpha)\sigma}}\int_{\mathbb{R}^2}(I_{\alpha}*F(e^\sigma u))F(e^{\sigma}u). \end{align}\] thus, \[\begin{align} \tilde{J}'_u(\sigma) & = & e^{2\sigma}|\nabla u|_2^2+\frac{se^{2s\sigma}}{2}[u]_s^2-\frac{\Lambda}{e^{(2+\alpha)\sigma}}\int_{\mathbb{R}^2}(I_{\alpha}*F(e^{\sigma}u)\tilde{F}(e^{\sigma}u)\\ & = &e^{2\sigma}\left(|\nabla u|_2^2 +\frac{s}{2}e^{2(s-1)\sigma}[u]_s^2-\frac{\Lambda}{e^{(4+\alpha)\sigma}}\int_{\mathbb{R}^2}(I_{\alpha}*F(e^{\sigma}u)\tilde{F}(e^{\sigma}u)\right)\\ & = & e^{2\sigma}\phi(\sigma), \end{align}\] where \[\begin{align} \phi(\sigma) & := & |\nabla u|_2^2 +\frac{s}{2}e^{2(s-1)\sigma}[u]_s^2-\frac{\Lambda}{e^{(4+\alpha)\sigma}}\int_{\mathbb{R}^2}(I_{\alpha}*F(e^{\sigma}u)\tilde{F}(e^{\sigma}u)\\ & = & |\nabla u|_2^2 +\frac{s}{2}e^{2(s-1)\sigma}[u]_s^2-\Lambda \int_{\mathbb{R}^2}\left(I_{\alpha}*\frac{F(e^{\sigma }u)}{e^{(2+\frac{\alpha}{2})\sigma}}\right)\frac{\tilde{F}(e^{\sigma}u)}{e^{(2+\frac{\alpha}{2})\sigma}}dx. \end{align}\] Now, since for any fixed \(t\neq 0\) by [f2] and [f4], \[\zeta\mapsto \frac{F(\zeta t)}{\zeta^{2+\frac{\alpha}{2}}}\text{ is strictly increasing for } \zeta\in (0,\infty),\] \[\zeta\mapsto \frac{\tilde{F}(\zeta t)}{\zeta^{2+\frac{\alpha}{2}}}\text{ is non decreasing for } \zeta\in (0,\infty),\] and \(e^{\sigma}\in (0,\infty)\) for all \(\sigma\in \mathbb{R}\), we get \[\sigma\mapsto -\Lambda \int_{\mathbb{R}^2}\left(I_{\alpha}*\frac{F(e^{\sigma }u)}{e^{(2+\frac{\alpha}{2})\sigma}}\right)\frac{\tilde{F}(e^{\sigma}u)}{e^{(2+\frac{\alpha}{2})\sigma}}dx \text{ is strictly decreasing in }\mathbb{R}.\] Further, since \(s\in (0,1)\), it turns out that \(\phi\) is strictly decreasing in \(\mathbb{R}\). Thus, it can vanish at most once. Therefore, \(\tilde{J}_u\) can have at most one critical point.

Moreover, by continuity of the exponential function, we can always find \(\sigma_0\) such that \[|\nabla H(u,\sigma_0)|_2^2+[H(u,\sigma_0)]_s^2 = e^{2\sigma_0}|\nabla u|_2^2 +e^{2s\sigma_0}[u]_s^2 =4 K(a)\Rightarrow H(u,\sigma_0)\in B,\] then by Lemma 3 and 4, we get \[\tilde{J}_u(\sigma)\rightarrow 0 \text{ as } \sigma \rightarrow -\infty;\; \tilde{J}_u(\sigma)\rightarrow -\infty \text{ as } \sigma \rightarrow+\infty \text{ and } \tilde{J}_{u}(\sigma_0)>0.\] Thus, \(\tilde{J}_u\) must have at least one critical point.
Therefore, \(\tilde{J}_u\) has the following curvature with the unique critical point \(\sigma(u)\).

Clearly, \(\sigma(u)\) corresponds to the global maximum of \(\tilde{J}_u\). Further, since \[P(H(u,\sigma(u)) = \tilde{J}'_{u}(\sigma(u))=0,\] we get \(H(u,\sigma(u))\in \mathcal{P}(a)\). Moreover, if \(v\in \mathcal{P}(a)\), then \[\tilde{J}'_v(0) = P(H(v,0))= P(v)=0,\] thus \(\sigma(v)=0.\) ◻

Lemma 6. Assume that [f1]-[f2] and [f4] hold. Then \(\gamma(a)=m(a),\) where \[m(a):=\inf_{u\in\mathcal{P}(a)}J(u).\] Moreover, \[m(a)= \inf_{u\in\mathcal{P}(a)} \max_{\sigma\in\mathbb{R}}J(H(u,\sigma)).\]

Proof. We first observe that, by Lemma 5, if \(v\in\mathcal{P}(a)\), then the map \[\sigma\mapsto J(H(v,\sigma))\] attains its unique maximum at \(\sigma=0\). Indeed, \(H(v,0)=v\in\mathcal{P}(a)\). Therefore, \[J(v)=\max_{\sigma\in\mathbb{R}}J(H(v,\sigma)).\] Taking the infimum over \(v\in\mathcal{P}(a)\), we obtain \[m(a) = \inf_{v\in\mathcal{P}(a)}J(v) = \inf_{v\in\mathcal{P}(a)} \max_{\sigma\in\mathbb{R}}J(H(v,\sigma)).\] We now prove that \(\gamma(a)\le m(a)\). Let \(v\in\mathcal{P}(a)\) be arbitrary. By Lemma 3, we have \[J(H(v,\sigma))\to 0 \quad\text{as }\sigma\to-\infty\] and \[J(H(v,\sigma))\to-\infty \quad\text{as }\sigma\to+\infty.\] Moreover, the scaling \(H(v,\sigma)\) preserves the \(L^2\)-norm, and hence \[H(v,\sigma)\in\mathcal{S}_r(a) \quad\text{for every }\sigma\in\mathbb{R}.\] Thus, we may choose \(\sigma_1<0<\sigma_2\) such that \[H(v,\sigma_1)\in A \quad\text{and}\quad J(H(v,\sigma_2))<0.\] Define \[h(t):=H(v,(1-t)\sigma_1+t\sigma_2), \qquad t\in[0,1].\] Then \(h\in\Gamma\). Hence, by the definition of \(\gamma(a)\), \[\gamma(a) \le \max_{t\in[0,1]}J(h(t)).\] Since \(h(t)\) is only a part of the fiber curve \(\sigma\mapsto H(v,\sigma)\), we have \[\max_{t\in[0,1]}J(h(t)) \le \max_{\sigma\in\mathbb{R}}J(H(v,\sigma)).\] Using again that \(v\in\mathcal{P}(a)\), Lemma 5 gives \[\max_{\sigma\in\mathbb{R}}J(H(v,\sigma))=J(v).\] Therefore, \[\gamma(a)\le J(v).\] Taking the infimum over \(v\in\mathcal{P}(a)\), we obtain \[\gamma(a)\le \inf_{v\in\mathcal{P}(a)}J(v)=m(a).\] It remains to prove that \(\gamma(a)\ge m(a)\). Let \(h\in\Gamma\) be arbitrary. By the definition of \(\Gamma\), we have \[h(0)\in A \quad\text{and}\quad J(h(1))<0.\] By choosing \(K(a)>0\) sufficiently small in the definition of \(A\), the same estimates used in Lemma 3 imply that \[P(w)>0 \quad\text{for every } w\in A.\] Hence, \[P(h(0))>0.\] On the other hand, we claim that \[J(w)<0 \quad \Longrightarrow \quad P(w)<0.\] Indeed, let \(w\in\mathcal{S}_r(a)\) and consider \[\varphi_w(\sigma):=J(H(w,\sigma)).\] By Lemma 5, \(\varphi_w\) has a unique maximum point \(\sigma(w)\), and \[\varphi_w'(\sigma)=P(H(w,\sigma)).\] Moreover, \(\varphi_w(\sigma)\to0\) as \(\sigma\to-\infty\). If \(P(w)\ge0\), then \(0\) lies before or at the maximum point of \(\varphi_w\), and consequently \[J(w)=\varphi_w(0)\ge0,\] which contradicts \(J(w)<0\). Therefore \(P(w)<0\). Applying this to \(w=h(1)\), we get \[P(h(1))<0.\] Since \(h\) is continuous in \(\mathcal{S}_r(a)\) and \(P\) is continuous on \(\mathcal{S}_r(a)\), the map \[t\mapsto P(h(t))\] is continuous on \([0,1]\). Since \[P(h(0))>0 \quad\text{and}\quad P(h(1))<0,\] the intermediate value theorem gives \(t_0\in(0,1)\) such that \[P(h(t_0))=0.\] Since \(h(t_0)\in\mathcal{S}_r(a)\), we have \[h(t_0)\in\mathcal{P}(a).\] Therefore, \[\max_{t\in[0,1]}J(h(t)) \ge J(h(t_0)) \ge \inf_{v\in\mathcal{P}(a)}J(v) = m(a).\] Taking the infimum over \(h\in\Gamma\), we obtain \[\gamma(a)\ge m(a).\] Combining \[\gamma(a)\le m(a) \quad\text{and}\quad \gamma(a)\ge m(a),\] we conclude that \[\gamma(a)=m(a).\] ◻

3.2 Analysis of the Palais-Smale Sequence↩︎

In this section we will study the Palais- Smale sequence corresponding to \(\gamma(a)\). Consider \[\tilde{\gamma}(a):= \inf_{\tilde{h}=(\tilde{h}_1,\tilde{h}_2)\in \tilde{\Gamma}}\max_{t\in [0,1]}J(H(\tilde{h}_1(t),\tilde{h}_2(t))),\] where \[\tilde{\Gamma} =\{\tilde{h}\in C([0,1], \mathcal{S}_r(a)\times \mathbb{R}): \tilde{h}(0)\in A\times \{0\} \text{ and } \tilde{h}(1)=(\tilde{h}_1(1),0) \text{ with } J(\tilde{h}_1(1))<0\}.\] Clearly, following the proof of [15] one can see that \(\tilde{\gamma}(a)=\gamma(a)\). Hence, using the Ekeland variational principle as explained in [24] we get sequences \(\{u_n\}\subset \mathcal{S}_r(a)\) and \(\{\lambda_n\}\subset \mathbb{R}\) such that as \(n\rightarrow +\infty\), \[\label{eq95PSS} \left\{\begin{array}{ll} J(u_n) \rightarrow \gamma(a)=m(a) & \text{ in } \mathbb{R}, \\ J'(u_n)+\lambda_n u_n \rightarrow 0 & \text{ in } H^{-1}(\mathbb{R}^2),\\ P(u_n) \rightarrow 0 & \text{ in }\mathbb{R}. \end{array} \right\}\tag{14}\] In the following, we study whether the value \(m(a)\) is achieved. To this end, we first establish some properties of the \((PS)\) sequence \(\{u_n\}\).

Lemma 7. Assume that [f1]-[f2] holds. Let \(u\in \mathcal{S}_{r}(a)\). Then the \((PS)\) sequence \(\{u_{n}\}\) of \(J\) is bounded in \(H^{1}(\mathbb{R}^{2})\).

Proof. For simplicity, set \(A_n:=|\nabla u_n|^2_2\,dx ;\) \(B_n:=[u_n]^2_s;\) \[C_n:=\int_{\mathbb{R}^2}(I_\alpha*F(u_n))F(u_n)\,dx; \text{ and } D_n:=\int_{\mathbb{R}^2}(I_\alpha*F(u_n))f(u_n)u_n\,dx.\] Then 14 become \[\label{Jn95mixed95expand} \frac{1}{2} A_n+\frac{1}{4} B_n- \frac{\Lambda}{2} C_n=\gamma(a),\tag{15}\] and \[\label{Pn95mixed95expand} A_n+\frac{s}{2}B_n+\Lambda\left(\frac{2+\alpha}{2}\right)C_n-\Lambda D_n=o_n(1).\tag{16}\] Now multiply 15 by \((2+\alpha)\) and add the result to 16 . We obtain \[\label{combine195mixed} \left(2+\frac{\alpha}{2}\right)A_n +\left(\frac{2+\alpha+2s}{4}\right)B_n - \Lambda D_n = (2+\alpha)\gamma(a)+o_n(1).\tag{17}\] On the other hand, by assumption [f2], there exists \(\theta>2+\frac{\alpha}{2}\) such that \[0<\theta F(t)\le f(t)t \qquad \text{for all } t\neq 0.\] Hence, \[D_n = \int_{\mathbb{R}^2}(I_\alpha*F(u_n))f(u_n)u_n\,dx \ge \theta\int_{\mathbb{R}^2}(I_\alpha*F(u_n))F(u_n)\,dx = \theta C_n.\] Using this estimate in 17 , we get \[\label{combine295mixed} \left(2+\frac{\alpha}{2}\right)A_n +\left(\frac{2+\alpha+2s}{4}\right)B_n -\Lambda\theta C_n \ge (2+\alpha)\gamma(a)+o_n(1).\tag{18}\] Next, from 15 , we can express \(C_n\) as \[\label{Cn95expand95mixed} C_n=\frac{1}{\Lambda}\left(A_n+\frac{1}{2} B_n-2\gamma(a)\right)+o_n(1).\tag{19}\] Substituting 19 into 18 , we obtain \[\begin{align} &(2+\tfrac{\alpha}{2})A_n+\left(\frac{2+\alpha+2s}{4}\right)B_n -\theta\Big(A_n+\frac{1}{2} B_n-2\gamma(a)+o_n(1)\Big) \\ &\ge (2+\alpha)\gamma(a)+o_n(1). \end{align}\] After rearranging, this yields \[\label{boundAB95mixed} \left(\theta-2-\frac{\alpha}{2}\right)A_n + \left(\frac{2\theta-2-\alpha-2s}{4}\right)B_n \le \bigl(2\theta-2-\alpha\bigr)\gamma(a)+o_n(1).\tag{20}\] Since \(\theta>2+\frac{\alpha}{2},\) we have \(2\theta-2-\alpha-2s>0.\) Therefore, both coefficients on the left-hand side of 20 are positive. It follows that \(\{A_n\}\), \(\{B_n\}\), and \(\{C_n\}\) are bounded, that is, \[\int_{\mathbb{R}^2}|\nabla u_n|^2\,dx \le C, \qquad [u_n]_s^2\le C, \qquad \int_{\mathbb{R}^2}(I_\alpha*F(u_n))F(u_n)\,dx\leq C\] for some constant \(C>0\) independent of \(n\).

Finally, since \(u_n\in \mathcal{S}_r(a)\), we also have \(|u_n|_2^2=a^2.\) Hence, \[\|u_n\|^2 = |\nabla u_n|_2^2 +\frac{[u_n]_s^2}{2}+|u_n|_2^2\] is bounded uniformly in \(n\). Therefore, \(\{u_n\}\) is bounded in \(H^1(\mathbb{R}^2)\). ◻

By Proposition 4, we have the following Lemma.

Lemma 8. Let \(\{u_n\}\subset S_r(a)\) be a sequence satisfying \[\limsup_{n\to +\infty}\gamma \|u_n\|^2<(2+\alpha)\pi.\] Then there exist \(t>1\), sufficiently close to \(1\), and a constant \(C>0\) such that \[\int_{\mathbb{R}^{2}}\bigl(e^{\gamma |u_n|^{2}}-1\bigr)^{t}\,dx\le C, \qquad \forall ~~n\in\mathbb{N}.\]

Proof. Since \[\limsup_{n\to +\infty}\gamma \|u_n\|^{2}<(2+\alpha)\pi,\] there exist \(\varepsilon>0\) and \(n_{0}\in\mathbb{N}\) such that \[\gamma \|u_n\|^{2}\le (2+\alpha)\pi-\varepsilon, \qquad \forall~~ n\ge n_{0}.\] Choose \(t>1\), close enough to \(1\), so that \[t\bigl((2+\alpha)\pi-\varepsilon\bigr)<(2+\alpha)\pi.\] Then, for every \(n\ge n_{0}\), \[t\gamma \|u_n\|^{2}<(2+\alpha)\pi.\] Now define \[v_n:=\frac{u_n}{\|u_n\|}.\] Then \(\|v_n\|=1\), and hence, by the \(ii)\) of Proposition 4, there exists a constant \(C_{1}>0\) such that \[\int_{\mathbb{R}^{2}} \left( e^{\,t\gamma \|u_n\|^{2}|v_n|^{2}}-1 \right)\,dx = \int_{\mathbb{R}^{2}} \left( e^{\,t\gamma |u_n|^{2}}-1 \right)\,dx \le C_{1}, \qquad \forall~ n\ge n_{0}.\] Since for every \(t>1\), such that \[\bigl(e^{\eta}-1\bigr)^{t}\le C_t\bigl(e^{t\eta}-1\bigr), \qquad \forall~ \eta\ge0,\] Hence, \[\int_{\mathbb{R}^{2}}\bigl(e^{\gamma |u_n|^{2}}-1\bigr)^{t}\,dx \le \int_{\mathbb{R}^{2}}\bigl(e^{t\gamma |u_n|^{2}}-1\bigr)\,dx \le C_1C_{t}=C'_t, \qquad \forall~ n\ge n_{0}.\] Finally, enlarging the constant if necessary to cover the finitely many indices \(n<n_{0}\), we obtain \[\int_{\mathbb{R}^{2}}\bigl(e^{\gamma |u_n|^{2}}-1\bigr)^{t}\,dx\le C, \qquad \forall~ n\in\mathbb{N},\] where the lemma follows by fixing a constant \[C = \max \left\{ C'_t,\; \int_{\mathbb{R}^2} \left(e^{\gamma |u_1|^2} - 1\right)^t \, dx,\; \ldots,\; \int_{\mathbb{R}^2} \left(e^{\gamma |u_{n_0}|^2} - 1\right)^t \, dx \right\}.\] This completes the proof. ◻

Lemma 9. Assume that [f1]-[f2] hold. Let \(\{u_{n}\}\subset \mathcal{S}_{r}(a)\) with \[\label{ene} \begin{align}\displaystyle \limsup_{n \rightarrow +\infty} \gamma \| u_n\|^{2} < (2+\alpha)\pi. \end{align}\qquad{(7)}\] If \(u_{n} \rightharpoonup u\) in \(H^{1}_{rad}(\mathbb{R}^{2})\) and \(u_n(x) \rightarrow u(x)\) a.e. in \(\mathbb{R}^2\), then \[\begin{align} \begin{aligned}\displaystyle |u_n|^{q}(e^{\gamma |u_n(x)|^{2}}-1) \rightarrow |u|^{q}(e^{\gamma |u(x)|^{2}}-1) \,\, in \,\, L^{1}(\mathbb{R}^{2}). \end{aligned} \end{align}\]

Proof. Setting \[\begin{align} \begin{aligned}\displaystyle h_n(x)=e^{\gamma |u_n|^{2}}-1. \end{aligned} \end{align}\] By ?? , there exists \(t>1\) close to \(1\) such that \[\begin{align} \begin{aligned}\displaystyle t\gamma \|u_{n}\|^{2}\leq(2+\alpha)\pi. \end{aligned} \end{align}\] Thus, by Lemma 8, we know that \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}\Big(e^{\gamma |u_{n}|^{2}}-1\Big)^{t}dx& \leq \int_{\mathbb{R}^{2}}\Big(e^{t\gamma |u_{n}|^{2}}-1\Big)dx<C, \end{aligned} \end{align}\] where \(C=C(t,a,\gamma)>0\). Then, \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}} \Big(h_n(x)\Big)^{t}dx=\int_{\mathbb{R}^{2}}\Big(e^{\gamma |u_{n}|^{2}}-1\Big)^{t}dx<C, \end{aligned} \end{align}\] which implies that \[\begin{align} \begin{aligned}\displaystyle h_{n} \in L^{t}(\mathbb{R}^{2}) \quad and \quad \sup_{n \in \mathbb{N}}|h_n|_{t}<+\infty. \end{aligned} \end{align}\] Therefore, \(\{h_{n}\}\) is a bounded sequence in \(L^{t}(\mathbb{R}^{2})\). By \(u_n \rightharpoonup u\) in \(H^{1}_{rad}(\mathbb{R}^{2})\), we know that \(u_n \rightarrow u\) a.e. in \(\mathbb{R}^2.\) Thus, using [36], we obtain that \[\label{sNewlimit} h_n \rightharpoonup h=e^{\gamma |u|^{2}}-1, \,\, in \,\, L^{t}(\mathbb{R}^{2}).\tag{21}\] Now, we show that \[\label{ssubimp} \begin{align}\displaystyle |u_{n}|^{q}\rightarrow |u|^{q} \;\;in \;\;L^{t^{\prime}}(\mathbb{R}^{2}), \end{align}\tag{22}\] where \(t^{\prime}=\frac{t}{t-1}\). Then, by the compact embedding \(H^{1}_{rad}(\mathbb{R}^{2}) \hookrightarrow L^{qt'}(\mathbb{R}^{2})\), we have \[\begin{align} \begin{aligned}\displaystyle u_{n} \rightarrow u \quad L^{qt^{\prime}}(\mathbb{R}^{2}). \end{aligned} \end{align}\] Hence, we get 22 . Together 21 with 22 , we know \[\begin{align} \begin{aligned}\displaystyle |u_n|^{q}(e^{\gamma |u_n(x)|^{2}}-1) \rightarrow |u|^{q}(e^{\gamma |u(x)|^{2}}-1) \,\, in \,\, L^{1}(\mathbb{R}^{2}). \end{aligned} \end{align}\] This completes the proof. ◻

By Lemma 9, we have the following two important corollaries.

Corollary 2. Assume that [f1]-[f2] hold. Let \(\{u_{n}\}\subset \mathcal{S}_{r}(a)\) with \[\begin{align}\displaystyle \limsup_{n \rightarrow +\infty} \gamma \| u_n\|^{2} < (2+\alpha)\pi. \end{align}\] If \(u_{n} \rightharpoonup u\) in \(H^{1}_{rad}(\mathbb{R}^{2})\) and \(u_{n}(x) \rightarrow u(x)\) a.e in \(\mathbb{R}^2\), then \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))f(u_{n})\psi dx \rightarrow \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))f(u)\psi dx, \;as \;n\rightarrow\infty, \end{aligned} \end{align}\] for any \(\psi\in C_{0}^{\infty}(\mathbb{R}^{2})\).

Proof. As we know from, [37], \[\label{choq} \begin{align}\displaystyle |I_{\alpha}\ast F(u_{n})|_{\infty}\leq C. \end{align}\tag{23}\] Hence, for any \(\psi\in C_{0}^{\infty}(\mathbb{R}^{2})\), we have \[\begin{align} \begin{aligned}\displaystyle |(I_{\alpha}\ast F(u_{n}))f(u_{n})\psi|\leq C|f(u_{n})||\psi|\leq \varepsilon|u_{n}|^{\tau}|\psi| +C|u_{n}|^{q-1}|\psi|(e^{\gamma |u_{n}|^{2}}-1). \end{aligned} \end{align}\] Let \(U=\text{supp }(\psi)\). Then, we obtain \[\begin{align} \begin{aligned}\displaystyle \int_{U}|u_{n}|^{\tau}|\psi|dx\rightarrow\int_{U}|u|^{\tau}|\psi|dx, \;as \;n\rightarrow\infty, \end{aligned} \end{align}\] and \[\begin{align} \begin{aligned}\displaystyle \int_{U}|u_{n}|^{q-1}|\psi|(e^{\gamma |u_{n}|^{2}}-1)dx\rightarrow\int_{U}|u|^{q-1}|\psi|(e^{\gamma |u|^{2}}-1)dx, \;as \;n\rightarrow\infty. \end{aligned} \end{align}\] Now, applying a variant of the Lebesgue Dominated Convergence Theorem, we can deduce that \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))f(u_{n})\psi dx \rightarrow \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))f(u)\psi dx, \;as \;n\rightarrow\infty, \end{aligned} \end{align}\] which completes the proof. ◻

Corollary 3. Assume that [f1]-[f2] hold. Let \(\{u_{n}\}\subset \mathcal{S}_{r}(a)\) with \[\label{eneb} \begin{align}\displaystyle \limsup_{n \rightarrow +\infty} \gamma \| u_n\|^{2} < (2+\alpha)\pi. \end{align}\qquad{(8)}\] If \(u_{n} \rightharpoonup u\) in \(H^{1}_{rad}(\mathbb{R}^{2})\) and \(u_{n}(x) \rightarrow u(x)\) a.e in \(\mathbb{R}^2\), then \[\begin{align} \begin{aligned}\displaystyle \lim_{n\to+\infty}\int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))F(u_{n})dx \rightarrow \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))F(u)dx, \end{aligned} \end{align}\] and \[\begin{align} \begin{aligned}\displaystyle \lim_{n\to+\infty}\int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))f(u_{n})u_{n}\,dx \rightarrow \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))f(u)u\,dx. \end{aligned} \end{align}\]

Proof. From 23 , we have \[\begin{align} \begin{aligned}\displaystyle |I_{\alpha}\ast F(u_{n})|_{\infty}\leq C. \end{aligned} \end{align}\] By ?? and ?? , we have \[\begin{align} \begin{aligned}\displaystyle |F(u_{n})|\leq\varepsilon|u_{n}|^{\tau+1}+C|u_{n}|^{q}(e^{\gamma |u_{n}|^{2}}-1)\,\, \text{ for all }\, u_{n} \in H^{1}(\mathbb{R}^{2}), \end{aligned} \end{align}\] where \(\gamma>\gamma_{0}\), \(\tau>3\) and \(q>2\). Hence, we have \[\begin{align} \begin{aligned}\displaystyle |(I_{\alpha}\ast F(u_{n}))F(u_{n})|\leq C|F(u_{n})|\leq\varepsilon C|u_{n}|^{\tau+1}+C|u_{n}|^{q}(e^{\gamma |u_{n}|^{2}}-1). \end{aligned} \end{align}\] By Lemma 9, we know \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}|u_n|^{q}(e^{\gamma |u_n(x)|^{2}}-1)dx \rightarrow \int_{\mathbb{R}^{2}} |u|^{q}(e^{\gamma |u(x)|^{2}}-1)dx \,\, as \,\, n\rightarrow\infty. \end{aligned} \end{align}\] By the compact embedding \(H^{1}_{rad}(\mathbb{R}^{2}) \hookrightarrow L^{p}(\mathbb{R}^{2})\), for \(p>2\), we have \[\begin{align} \begin{aligned}\displaystyle u_{n}\rightarrow u \quad in \;\;L^{p}(\mathbb{R}^2). \end{aligned} \end{align}\] Now, applying a variant of the Lebesgue Dominated Convergence Theorem, we can deduce that \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))F(u_{n})dx \rightarrow \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))F(u)dx \,\, as \,\, n\rightarrow\infty. \end{aligned} \end{align}\] A similar argument works to show that \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))f(u_{n})u_{n}\,dx \rightarrow \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))f(u)u\,dx \,\, as \,\, n\rightarrow\infty, \end{aligned} \end{align}\] which completes the proof. ◻

Lemma 10. Assume that [f1]-[f2] hold. Let \(\{u_n\}\subset \mathcal{S}_r(a)\) and \(\{\lambda_n\}\subset\mathbb{R}\) be the sequences satisfying 14 . Suppose that \(\{u_n\}\) is bounded in \(H^1(\mathbb{R})\), then \(\{\lambda_n\}\) is bounded with \[\label{lambda-mixed-formula}\liminf_{n\to\infty}\lambda_n = \liminf_{n\to\infty} \left[ \left(\frac{s-1}{2}\right)\frac{1}{a^2}[u_n]_s^2 + \Lambda \left(\frac{2+\alpha}{2a^2}\right) \int_{\mathbb{R}^2}(I_\alpha *F(u_n))F(u_n)\,dx \right].\qquad{(9)}\]

Proof. Since \(\{u_n\}\) is bounded in \(H^{1}(\mathbb{R}^2)\) and 14 holds, the sequence \(\{\lambda_n\}\) is bounded. Also, since \(|u_n|_2^2=a^2\), we get \[\label{test-un} \int_{\mathbb{R}^2} |\nabla u_n|^2\,dx +\frac{1}{2}[u_n]_s^2 +\lambda_n a^2 = \Lambda\int_{\mathbb{R}^2}(I_\alpha *F(u_n))f(u_n)u_n\,dx +o_n(1).\tag{24}\] Hence, \[\label{lambda-step1} \lambda_n a^2 = -\int_{\mathbb{R}^2} |\nabla u_n|^2\,dx -\frac{1}{2}[u_n]_s^2 + \Lambda \int_{\mathbb{R}^2}(I_\alpha *F(u_n))f(u_n)u_n\,dx +o_n(1).\tag{25}\] On the other hand, from 14 we have \[\label{pohozaev-rewrite} \int_{\mathbb{R}^2}(I_\alpha *F(u_n))f(u_n)u_n\,dx = \int_{\mathbb{R}^2} |\nabla u_n|^2\,dx +\frac{s}{2} [u_n]_s^2 +\Lambda \left(\frac{2+\alpha}{2}\right)\int_{\mathbb{R}^2}(I_\alpha *F(u_n))F(u_n)\,dx +o_n(1).\tag{26}\] Substituting 26 into 25 , we obtain \[\lambda_n a^2 = \left(\frac{s-1}{2}\right)[u_n]_s^2 +\Lambda \left(\frac{2+\alpha}{2}\right)\int_{\mathbb{R}^2}(I_\alpha *F(u_n))F(u_n)\,dx +o_n(1).\] Dividing by \(a^2\) yields ?? . The \(\liminf\) identity follows immediately. ◻

4 Existence Results↩︎

4.1 Existence result for the subcritical case↩︎

In this section, we assume that \(f\) has subcritical growth and restrict our analysis to \(H^{1}_{rad}(\mathbb{R}^{2})\).

Proof of Theorem 1. Let \(\{u_n\}\) be the Palais Smale sequence constructed in section 3.2 that satisfies 14 . First, we show that \(u_n \rightharpoonup u\) in \(H^{1}_{rad}(\mathbb{R}^{2})\), with \(u \neq 0\).

By Lemma 7, the sequence \(\{u_n\}\) is bounded in \(H^{1}_{rad}(\mathbb{R}^{2})\), hence up to a subsequence, \[u_n \rightharpoonup u \quad \text{in } H^{1}_{rad}(\mathbb{R}^{2}).\] For \(\gamma > 0\) sufficiently small and since \(\{u_n\}\) is bounded, we have \[\limsup_{n \to +\infty} \gamma \|u_n\|^{2} < (2+\alpha)\pi.\] Then, by Corollary 3, it follows that \[\label{limitaa-mixed} \lim_{n \to +\infty} \int_{\mathbb{R}^2}(I_{\alpha}\ast F(u_{n}))f(u_{n})u_{n}\,dx = \int_{\mathbb{R}^2}(I_{\alpha}\ast F(u))f(u)u\,dx,\tag{27}\] and \[\label{limitab-mixed} \lim_{n \to +\infty} \int_{\mathbb{R}^2}(I_{\alpha}\ast F(u_{n}))F(u_{n})\,dx = \int_{\mathbb{R}^2}(I_{\alpha}\ast F(u))F(u)\,dx.\tag{28}\] We claim that \(u \neq 0\). Otherwise, by Corollary 3, we have \[\begin{align} \begin{aligned}\displaystyle \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))F(u_{n})dx = \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))f(u_{n})u_{n}\,dx=0, \end{aligned} \end{align}\] Then, by Lemma 10 and [f2], we obtain, \(\lambda_n\geq 0\), since \[\begin{align} \liminf_{n\to\infty}\lambda_n &= \frac{1}{a^{2}}\left(\frac{s-1}{2}\right)[u_n]_s^2 +\Lambda\left(\frac{2+\alpha}{2a^{2}}\right) \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u_{n}))F(u_{n})\,dx \\ &= \frac{1}{a^{2}}\left(\frac{s-1}{2}\right)[u]_s^2 +\Lambda\left(\frac{2+\alpha}{2a^{2}}\right) \int_{\mathbb{R}^{2}}(I_{\alpha}\ast F(u))F(u)\,dx \\ & \geq 0 \text{ for sufficiently large }\Lambda>0. \end{align}\] On the other hand, testing the following equation \[\mathcal{L} u_n + \lambda_n u_n = \Lambda(I_{\alpha}\ast F(u_n))f(u_n)+o_n(1)\] by \(u_n\), then we obtain \[|\nabla u_n|_2^2 + \frac{1}{2}[u_n]_s^2 + \lambda_n a^2 = \Lambda\int_{\mathbb{R}^2}(I_{\alpha}\ast F(u_n))f(u_n)u_n\,dx + o_n(1),\] leads to \[\label{m2-mixed} -\lambda_n a^2 = |\nabla u_n|_2^2 +\frac{1}{2} [u_n]_s^2 + o_n(1).\tag{29}\] From this, \[0 \geq -\liminf_{n\to\infty}\lambda_n a^2= \limsup_{n\to\infty} (-\lambda_n) a^2 = \limsup_{n\to\infty} \big(|\nabla u_n|_2^2 + \frac{1}{2}[u_n]_s^2\big) \geq \liminf_{n\to\infty} \big(|\nabla u_n|_2^2 + \frac{1}{2}[u_n]_s^2\big)\geq 0,\] which implies \[|\nabla u_n|_2^2 + \frac{1}{2}[u_n]_s^2 \to 0,\] a contradiction with \(\gamma(a)>0\). Therefore, \(u \neq 0\).

Next, we show that \(\lambda>0\). By Lemma 10, [f2], and \(u\neq 0\) there exists a bounded sequence \(\{\lambda_n\}\). Thus, up to a subsequence, \(\lambda_n \to \lambda>0, ~~\text{as}~~n \to \infty.\) Then, by Corollary 2, we have \[\label{5eq1} \mathcal{L} u + \lambda u = \Lambda(I_{\alpha}\ast F(u))f(u) \quad \text{in } \mathbb{R}^{2}.\tag{30}\] Moreover, we deduce that \(P(u)=0\). Now, we obtain that \(u_n\rightharpoonup u \neq 0.\) Then, we now prove the strong convergence \(u_n \to u\) in \(H^{1}_{rad}(\mathbb{R}^{2})\). The proof is divided into the two steps.

Step 1. We show that \[\lim_{n\to\infty} \big(|\nabla u_n|_2^2 + [u_n]_s^2\big) = |\nabla u|_2^2 + [u]_s^2.\] By \({P}(u)=0\) and 14 with 27 and 28 , we obtain \[P(u_n)-P(u)=o_n(1),\] which implies \[|\nabla u_n|_2^2 + [u_n]_s^2 \to |\nabla u|_2^2 + [u]_s^2.\]

Step 2. We show that \(|u|_2 = a\).

Combining Corollary 3, Lemma 10, and Pohožaev identity [Pohozaev], we obtain \[\lambda a^2 = \lambda |u|_2^2.\] Since \(\lambda>0\), it follows that \[|u|_2 = a.\] Therefore, \[u_n \to u \quad \text{in } H^{1}_{rad}(\mathbb{R}^{2}).\] Finally, by Lemma 6, we conclude that \(u\) is a normalized ground state solution of the problem 1 . ◻

4.2 Existence result for the critical growth↩︎

In this subsection, we treat the case of critical exponential growth in the radial space \(H^1_{\mathrm{rad}}(\mathbb{R}^2)\). We first prove three auxiliary lemmas, namely Lemmas 11, 12, and 13. These lemmas provide a suitable upper bound for the minimax level \(\gamma(a)\), which is the key point in showing that the associated Palais-Smale sequence lies below the Trudinger-Moser critical threshold. This allows us to apply the compactness analysis developed in Subsection 3.2 and complete the proof of Theorem 2.

Lemma 11. Assume that [f3] holds. Then \[\lim_{\mu \rightarrow +\infty}\gamma(a)=0.\]

Proof. Fix \(u_0\in \mathcal{S}_r(a)\) and consider the path \[h_0(t)=H\bigl(u_0,\sigma_t)\in \Gamma, \qquad \sigma_1<0,~~~\sigma_2>0,~~~~t\in[0,1],\] where \[\sigma_t:=(1-t)\sigma_1+t\sigma_2.\] By the definition of the minimax level \(\gamma(a)\), we have \[\gamma(a)\le \max_{t\in[0,1]} J(h_0(t)).\] Now, using the scaling, we deduce that \[\begin{align} J(h_0(t)) &= \frac{e^{2\sigma_t}}{2}|\nabla u_0|_2^2 +\frac{e^{2s\sigma_t}}{4}[u_0]_s^2 -\frac{\Lambda}{2e^{(2+\alpha)\sigma_t}} \int_{\mathbb{R}^2}(I_\alpha*F(e^{\sigma_t}u_0))F(e^{\sigma_t}u_0)\,dx. \end{align}\] By assumption [f3], there exists \(\tilde{\sigma}>2+\frac{\alpha}{2}\) such that \[F(t)\geq \mu |t|^{\tilde{\sigma}}\ge \frac{\mu}{\tilde{\sigma}}|t|^{\tilde{\sigma}} \qquad \text{for all } t\in\mathbb{R}.\] Therefore, \[F(e^{\sigma_t}u_0)\ge \frac{\mu}{\tilde{\sigma}}e^{\tilde{\sigma}\sigma_t}|u_0|^{\tilde{\sigma}}.\] Substituting this into the Choquard term, we get \[\begin{align} &\int_{\mathbb{R}^2}(I_\alpha*F(e^{\sigma_t}u_0))F(e^{\sigma_t}u_0)\,dx \ge \frac{\mu}{\tilde{\sigma}}e^{\tilde{\sigma}\sigma_t} \int_{\mathbb{R}^2}(I_\alpha*F(e^{\sigma_t}u_0))|u_0|^{\tilde{\sigma}}\,dx. \end{align}\] Using the Hardy-Littlewood-Sobolev inequality, we infer that \[\int_{\mathbb{R}^2}(I_\alpha*F(e^{\sigma_t}u_0))F(e^{\sigma_t}u_0)\,dx \ge \frac{C\mu}{\sigma}e^{2\tilde{\sigma}\sigma_t} |u_0|_{\frac{8\tilde{\sigma}}{2+\alpha}}^{4\tilde{\sigma}},\] for some constant \(C>0\) independent of \(\mu\) and \(\sigma_t\). Consequently, \[\begin{align} J(h_0(t)) &\le \frac{e^{2\sigma_t}}{2}|\nabla u_0|_2^2 +\frac{e^{2s\sigma_t}}{4}[u_0]_s^2 -\frac{\Lambda C\mu}{\tilde{\sigma}}e^{(2\tilde{\sigma}-2-\alpha)\sigma_t} |u_0|_{\frac{8\tilde{\sigma}}{2+\alpha}}^{4\tilde{\sigma}}. \end{align}\] By the definition of \(\sigma_t,\) we have \[\sigma_t\in [\min\{\sigma_1,\sigma_2\},\,\max\{\sigma_1,\sigma_2\}] \qquad \text{for all } t\in[0,1].\] Now we may estimate \[\begin{align} \gamma(a) &\le \max_{t\in[0,1]}J(h_0(t)) \\ &\le \max_{\tau\in\mathbb{R}} \left\{ \frac{e^{2\tau}}{2}|\nabla u_0|_2^2 +\frac{e^{2s\tau}}{4}[u_0]_s^2 -\frac{\Lambda C\mu}{\sigma}e^{(2\tilde{\sigma}-2-\alpha)\tau} |u_0|_{\frac{8\tilde{\sigma}}{2+\alpha}}^{4\tilde{\sigma}} \right\}. \end{align}\] Now set \(r=e^\tau>0.\) Then \[\begin{align} \gamma(a) \le \max_{r>0} \left\{ \frac{r^2}{2}|\nabla u_0|_2^2 +\frac{r^{2s}}{4}[u_0]_s^2 -\frac{\Lambda C\mu}{\sigma}r^{2\tilde{\sigma}-2-\alpha} |u_0|_{\frac{8\tilde{\sigma}}{2+\alpha}}^{4\tilde{\sigma}} \right\}. \end{align}\] Since \(0<s<1\) and \(\tilde{\sigma}>2+\frac{\alpha}{2}\). Thus, \[\gamma(a)\le C_1\left(\frac{1}{\mu}\right)^{\frac{2}{2\tilde{\sigma}-4-\alpha}}\] for some constant \(C_1>0\) independent of \(\mu\). In particular, \[\gamma(a)\to 0 \qquad \text{as } \mu\to +\infty.\] This completes the proof. ◻

Lemma 12. Assume that [f3] holds. Let \(\{u_{n}\}\subset \mathcal{S}_{r}(a)\) be the sequence satisfying 14 . Then \[\limsup_{n \rightarrow +\infty} \int_{\mathbb{R}^2}(I_{\alpha}\ast F(u_{n}))F(u_{n})\,dx \leq \frac{1}{\Lambda}\left(\frac{4}{2\theta-4-\alpha}\gamma(a)\right).\]

Proof. Set \(A_n:=|\nabla u_n|^2_2 ;\) \(B_n:=[u_n]^2_s;\) and \[N_n:=\int_{\mathbb{R}^2}(I_\alpha*F(u_n))F(u_n)\,dx, \qquad M_n:=\int_{\mathbb{R}^2}(I_\alpha*F(u_n))f(u_n)u_n\,dx.\] Since \(\{u_n\}\) satisfies 14 , we have \[J(u_n)=\gamma(a)+o_n(1) \qquad \text{and} \qquad P(u_n)=o_n(1).\] Also, we may write \(J(u_n)= \gamma(a)+ {P}(u_n)\). From the definition of \(J\), it follows that \[\frac{1}{2} A_n+\frac{1}{4} B_n-\frac{\Lambda}{2} N_n=\gamma(a)+o_n(1),\] that is, \[\label{eq1-newlem1} A_n+\frac{1}{2} B_n= \Lambda N_n+2\gamma(a)+o_n(1).\tag{31}\] On the other hand, from the Pohožaev identity [Pohozaev], we obtain \[A_n+\frac{s}{2}B_n+\Lambda\left(\frac{2+\alpha}{2}\right)N_n- \Lambda M_n=o_n(1).\] Adding this relation to \((2+\alpha)J(u_n)=(2+\alpha)\gamma(a)+o_n(1)\), we get \[\frac{4+\alpha}{2}A_n+\frac{2+\alpha+2s}{4}B_n- \Lambda M_n =(2+\alpha)\gamma(a)+o_n(1).\] Using [f2], namely \[f(t)t\ge \theta F(t)\qquad \text{for all }t\in\mathbb{R},\] we infer that \[M_n\ge \theta N_n.\] Hence, \[\label{eq2-newlem1} (2+\alpha)\gamma(a)+o_n(1) \le \frac{4+\alpha}{2}A_n+\frac{2+\alpha+2s}{4}B_n- \Lambda \theta N_n.\tag{32}\] Now, by 31 , \[A_n= \Lambda N_n+2\gamma(a)-\frac{1}{2} B_n+o_n(1).\] Substituting this into 32 , we find \[\begin{align} (2+\alpha)\gamma(a)+o_n(1) &\le \frac{4+\alpha}{2}\left(\Lambda N_n+2\gamma(a)-\frac{1}{2} B_n\right) +\frac{2+\alpha+2s}{4}B_n-\theta N_n+o_n(1)\\ &= (4+\alpha)\gamma(a) +\left(\frac{4+\alpha}{2}-\Lambda \theta\right)\Lambda N_n +\frac{2s-2}{4}B_n +o_n(1). \end{align}\] Since \(0<s<1\), we have \[\frac{2s-2}{4}B_n\le 0.\] Therefore, \[(2+\alpha)\gamma(a)+o_n(1) \le (4+\alpha)\gamma(a)+\left(\frac{4+\alpha}{2}-\theta\right) \Lambda N_n+o_n(1),\] which yields \[\left(\theta-\frac{4+\alpha}{2}\right)\Lambda N_n \le 2\gamma(a)+o_n(1).\] Since \(\theta>2+\frac{\alpha}{2}\), we conclude that \[N_n\le \frac{1}{\Lambda}\left(\frac{4}{2\theta-4-\alpha}\gamma(a)\right)+o_n(1).\] Taking the limit superior as \(n\to+\infty\), we obtain \[\limsup_{n\to+\infty} \int_{\mathbb{R}^2}(I_\alpha*F(u_n))F(u_n)\,dx \le \frac{1}{\Lambda}\left(\frac{4}{2\theta-4-\alpha}\gamma(a)\right).\] This completes the proof. ◻

Lemma 13. Assume that [f3] holds. Let \(\{u_{n}\}\subset \mathcal{S}_{r}(a)\) be the sequence satisfying 14 . Then \[\limsup_{n \rightarrow +\infty} \left(|\nabla u_n|_2^2+\frac{[u_n]_s^2}{2}\right) \leq \left(\frac{4\theta-4-2\alpha}{2\theta-4-\alpha}\right)\gamma(a).\] Hence, there exists \(\mu^*>0\) such that \[\label{level-mixed} \limsup_{n \to +\infty} \left(|\nabla u_n|_2^2+\frac{[u_n]_s^2}{2}\right) < \frac{(2+\alpha)\pi}{\gamma_{0}}-a^{2}, \quad \text{for all } \mu \geq \mu^*.\qquad{(10)}\]

Proof. Using the notations \(A_n\), \(B_n\), and \(N_n\) introduced in the proof of Lemma 12, we proceed as follows. Since \(\{u_n\}\) satisfies 14 , we have \[J(u_n)=\gamma(a)+o_n(1).\] By the definition of \(J\), it follows that \[\label{eq-bound1} A_n+\frac{1}{2} B_n= \Lambda N_n+2\gamma(a)+o_n(1).\tag{33}\] Taking \(\limsup\) and using Lemma 12, we obtain \[\begin{align} \limsup_{n\to\infty}(A_n+\frac{B_n}{2}) &\le 2\gamma(a) +\Lambda \limsup_{n\to\infty} N_n\\ &\le 2\gamma(a) +\frac{4}{2\theta-4-\alpha}\gamma(a). \end{align}\] Therefore, \[\begin{align} \limsup_{n\to\infty}(A_n+\frac{B_n}{2}) &\le 2\gamma(a) + \frac{4}{2\theta-4-\alpha}\gamma(a)\\ &= \frac{4\theta-4-2\alpha}{2\theta-4-\alpha}\gamma(a). \end{align}\] The estimate ?? follows from the fact that \(\gamma(a)\to 0\) as \(\mu\to+\infty\), so for \(\mu\) sufficiently large, \[\frac{4\theta-4-2\alpha}{2\theta-4-\alpha}\gamma(a) < \frac{(2+\alpha)\pi}{\gamma_0}-a^2.\] This completes the proof. ◻

Proof of Theorem 2. By Lemma 13, we have \[\limsup_{n \to +\infty} \left( |\nabla u_{n}|_{2}^{2} + \frac{[u_n]_s^2}{2} \right) < \frac{(2+\alpha)\pi}{\gamma_{0}}-a^{2}, \quad \text{for any } \mu \geq \mu^*,\] which implies that \[\limsup_{n \to +\infty} \|u_n\|^{2} < \frac{(2+\alpha)\pi}{\gamma_{0}}, \quad \text{for any } \mu \geq \mu^*.\] Hence, for \(\gamma>\gamma_{0}\) sufficiently close to \(\gamma_{0}\), we deduce that \[\limsup_{n \rightarrow +\infty} \gamma \| u_n\|^{2} < (2+\alpha)\pi.\] Following the same arguments as in subsection 4.1, we conclude that, up to a subsequence, \[u_n \to u \quad \text{in } H^{1}_{rad}(\mathbb{R}^{2}).\] This completes the proof. ◻

5 Appendix: Regularity and Pohožaev Identity↩︎

In order to get the existence of the normalized solution, we have constructed the Pohožaev manifold using the following Pohožaev identity:

Theorem 6. Let \(u\in H^1(\mathbb{R}^2)\) be a weak solution to 1 , the it satisfies the following: \[\label{Pohozaev95identity} \left(\frac{1-s}{2}\right)[u]_s^2+\lambda | u |_2^2 = \left(\frac{2+\alpha}{2}\right)\mathcal{A}_{F}(u),\qquad{(11)}\] where \[\mathcal{A}_F(u):=\Lambda\int_{\mathbb{R}^2}(I_{\alpha}*F(u))F(u).\]

We will be following the work of [38] for the case of Choquard nonlinearity as done in [14]; and for that, we need any weak solution to be H\(\ddot{\text{o}}\)lder continuous. Let us start with studying the regularity of the solution.

Theorem 7. Let \(u\in H^1(\mathbb{R}^2)\) be a weak solution of: \[-\Delta u +(-\Delta)^s u +\lambda u = \Lambda(I_{\alpha}*F(u))f(u) \text{ in } \mathbb{R}^2,\] then \(u\in C^{1,\delta}_{loc}(\mathbb{R}^2)\), for all \(0<\delta<1\).

Proof. Due to the imbedding results, precisely [39], it is very well known that \(u\in L^t(\mathbb{R}^2)\) for all \(t\geq 2\). Next, we claim that \((I_{\alpha}*F(u))\in L^{\infty}(\mathbb{R}^2)\).
Now, for any \(x\in \mathbb{R}^2\), by ?? we have: \[\begin{align} (I_{\alpha}*F(u))(x) & = & \int_{\mathbb{R}^2} \frac{A_{\alpha}F(u(y))}{|x-y|^{2-\alpha}}dy = \int_{\mathbb{R}^2} \frac{A_{\alpha}F(u(x-y))}{|y|^{2-\alpha}}dy\\ & \leq & A_{\alpha}\varepsilon\int_{\mathbb{R}^2}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy + A_{\alpha}C_{\varepsilon} \int_{\mathbb{R}^2}\frac{|u(x-y)|^q\left(e^{\gamma|u(x-y)|^2}-1\right)}{|y|^{2-\alpha}}dy, \end{align}\] with \(\tau>3\) and \(q>2\). Denote \[I_1=\int_{\mathbb{R}^2}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy; \text{ and } I_2=\int_{\mathbb{R}^2}\frac{|u(x-y)|^q\left(e^{\gamma|u(x-y)|^2}-1\right)}{|y|^{2-\alpha}}dy.\] For any fixed \(\beta> \frac{2}{\alpha}>1\), we take \(p\in \left(1,\frac{2(\beta-1)}{\beta(2-\alpha)}\right)\). Then, denoting \(B_1\) to be the ball centered at origin, we have: \[I_1 = \int_{\mathbb{R}^2}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy=\int_{B_1}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy +\int_{\mathbb{R}^2\setminus B_1}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy,\] here, \[\int_{\mathbb{R}^2\setminus B_1}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy \leq \int_{\mathbb{R}^2\setminus B_1}|u(x-y)|^{\tau+1}dy \leq \int_{\mathbb{R}^2}|u(y)|^{\tau+1}dy<+\infty;\] and by H\(\ddot{\text{o}}\)lder’s inequality we have: \[\begin{align} \int_{B_1}\frac{|u(x-y)|^{\tau+1}}{|y|^{2-\alpha}}dy & \leq &\left(\int_{B_1}|u(x-y)|^{\beta(\tau+1)}\right)^{\frac{1}{\beta}}\left(\int_{B_1} \frac{dy}{|y|^{(2-\alpha)\frac{\beta}{\beta-1}}}\right)^{\frac{\beta-1}{\beta}} \\ & \le & K_1 \left( \int_{\mathbb{R}^2}|u|^{\beta(\tau+1)}\right)^{\frac{1}{\beta}}<+\infty, \end{align}\] for \(\beta>\frac{2}{\alpha}\). Now, using Proposition 4 and H\(\ddot{\text{o}}\)lder’s inequality, we estimate \(I_2\) as follows: \[\begin{align} I_2 & = & \int_{\mathbb{R}^2}\frac{|u(x-y)|^q\left(e^{\gamma|u(x-y)|^2}-1\right)}{|y|^{2-\alpha}}dy\\ & \leq & \left(\int_{\mathbb{R}^2}|e^{\gamma|u(x-y)|^2}-1|^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}} \left(\int_{\mathbb{R}^2}\frac{|u(x-y)|^{qp}}{|y|^{p(2-\alpha)}}dy\right)^{\frac{1}{p}}\\ & = & \left(\int_{\mathbb{R}^2}|e^{\gamma|u(x)|^2}-1|^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}} \left(\int_{\mathbb{R}^2}\frac{|u(x-y)|^{qp}}{|y|^{p(2-\alpha)}}dy\right)^{\frac{1}{p}} \le K_2 \left(\int_{\mathbb{R}^2}\frac{|u(x-y)|^{qp}}{|y|^{p(2-\alpha)}}dy\right)^{\frac{1}{p}}\\ & = & K_2 \left( \int_{B_1}\frac{|u(x-y)|^{qp}}{|y|^{p(2-\alpha)}}dy+\int_{\mathbb{R}^2\setminus B_1}\frac{|u(x-y)|^{qp}}{|y|^{p(2-\alpha)}}dy\right)^{\frac{1}{p}}\\ & \le & K_2 \left(\int_{B_1}\frac{|u(x-y)|^{pq}}{|y|^{p(2-\alpha)}}dy+\int_{\mathbb{R}^2} |u(x)|^{pq}dx \right)^{\frac{1}{p}}\\ & \le & K_3 \left(1+ \left(\int_{B_1}\frac{|u(x-y)|^{pq}}{|y|^{p(2-\alpha)}}\right)^{\frac{1}{p}}\right)\\ & \leq & K_3 \left(1+ \left(\int_{B_1}|u(x-y)|^{pq\beta}\right)^{\frac{1}{p\beta}}\left(\int_{B_1}\frac{dy}{|y|^{(2-\alpha)}\frac{p \beta}{\beta-1}}\right)^{\frac{(\beta-1)}{p\beta}}\right)<+\infty, \end{align}\] for \(p<\frac{2(\beta-1)}{\beta(2-\alpha)}\). Thus \((I_{\alpha}*F(u))\in L^{\infty}(\mathbb{R}^2)\). Now, let us reformulate our problem as: \[-\Delta u +\mathcal{L}_1 u +\lambda u = \Lambda(I_{\alpha}*F(u))f(u)-\mathcal{L}_2u;\] where \[\mathcal{L}_1 u(x) = C(2,s)P.V \int_{|x-y|\leq B}\frac{u(x)-u(y)}{|x-y|^{2+2s}}dy;\] and, \[\mathcal{L}_2u(x) =C(2,s)P.V \int_{|x-y|>B}\frac{u(x)-u(y)}{|x-y|^{2+2s}}dy,\] for some fixed \(B>0\). Define \(g(x)= \Lambda(I_{\alpha}*F(u))(x)f(u)(x)-\mathcal{L}_2(u)(x)\), then by Jensen’s inequality, above claim, ?? and the fact that \(u\in L^t (\mathbb{R}^2)\), for all \(t\ge 2\), we get \[\begin{align} \int_{\mathbb{R}^2}|g(x)|^t & \le & K_4\left(\Lambda\int_{\mathbb{R}^2}|(I_{\alpha}*F(u))f(u)|^t+\int_{\mathbb{R}^2} |\mathcal{L}_2(u)|^t\right)\\ & \le & K_5 \left(\Lambda\int_{\mathbb{R}^2}|f(u)|^t +\int_{\mathbb{R}^2}\left(\int_{|x-y|>B}\frac{|u(x)-u(y)|^t}{|x-y|^{t(2+2s)}}dy\right)dx\right)\\ & \le & K_6 \left(\varepsilon\Lambda\int_{\mathbb{R}^2}|u|^{t\tau}+C_{\varepsilon}\int_{\mathbb{R}^2}|u|^{(q-1)t}\left(e^{\gamma t |u|^2}-1\right)\right.\\ && \left.+2 \int_{\mathbb{R}^2}|u(x)|^t\left(\int_{|x-y|>B} \frac{dy}{|x-y|^{t(2+2s)}}\right)dx\right)\\ & \le & K_7 \left(\varepsilon\Lambda \int_{\mathbb{R}^2}|u(x)|^{t\tau}+C_\varepsilon\left(\int_{\mathbb{R^2}} |u|^{2t(q-1)} \right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^2}\left(e^{\gamma t |u|^2}-1\right)^2\right)^{\frac{1}{2}}\right.\\ && \left. +2\int_{\mathbb{R}}^2|u|^t\right)<+\infty \text{ for all } t\geq 2. \end{align}\] Thus \(g\in L^t(\mathbb{R}^2)\) for all \(t\ge 2\). Now, by [40] with \(\Omega_I\) being the ball of radius \(B\) centered at origin and a fixed bounded domain \(\Omega\), get \(u\in W^{2,t}_{loc}(\mathbb{R}^2)\) for all \(t\geq 2\). Further using Sobolev inequality [39], we conclude that \(u \in C^{1,\delta}_{\text{loc}}(\mathbb{R}^2)\), for all \(\delta \in (0,1)\). ◻

Moreover, we will need the following result in order to get our Pohožaev identity.

Proposition 8. Let \(u\) be a solution of 1 and \(\tilde{u}\) lies between \(u\) and \(u(.+he_j)\) for any \(h>0\) and unit vector \(e_j\). Then \(f(\tilde{u})\rightarrow f(u)\) in \(L^2(\mathbb{R}^2)\).

Proof. Now, since \(f\) is continuous and \(\tilde{u}\rightarrow u\) a.e. in \(\mathbb{R}^2\) as \(h\rightarrow 0\), we get: \[f(\tilde{u})\rightarrow f(u) \text{ a.e. in } \mathbb{R}^2.\] We will be using Vitali’s convergence theorem in order to prove the convergence. In that context, firstly, we need uniform integrability, that is, for any \(\eta>0\) we should be able to find a \(\delta>0\) independent of \(f(\tilde{u})\), such that: \[\sup_{|h|\leq h_0} \int_{E} |f(\tilde{u})|^2 <\eta \text{ whenever } |E|<\delta.\] Now, by ?? and the fact that \[|\tilde{u}(x)|\leq |u(x)|+|u(x+he_j)| \text{ for all } x\in \mathbb{R}^2,\] we have: \[\begin{align} |f(\tilde{u})|^{2r} & \leq & C_r \varepsilon^{2r} |\tilde{u}|^{2r\tau} +C_r'C_{\varepsilon}^{2r}|\tilde{u}|^{2r(q-1)}\left(e^{2r\gamma|\tilde{u}|^2}-1\right)\\ & \leq & C_r \varepsilon^{2r} |{u}+u(.+he_j)|^{2r\tau} + C_r'C_{\varepsilon}^{2r}|{u}+u(.+he_j)|^{2r(q-1)}\left(e^{2r\gamma|{u}|^2}e^{2r\gamma u(.+he_j)|^2}-1\right)\\ & \leq & C_r\varepsilon^{2r}C_1|u|^{2r\tau} +C_rC_1\varepsilon^{2r}|u(.+he_j)|^{2r\tau}\\ && +C_r'C_{\varepsilon}^{2r}\left(|u|^{2r(q-1)}+|u(.+he_j)|^{2r(q-1)}\right)\left(e^{2r\gamma|{u}|^2}e^{2r\gamma u(.+he_j)|^2}-1\right), \end{align}\] further since \((a-1)(b-1)+a-1+b-1=ab-1\), we get: \[\begin{align} |f(\tilde{u})|^{2r} & \leq & C_r\varepsilon^{2r}C_1|u|^{2r\tau} +C_rC_1\varepsilon^{2r}|u(.+he_j)|^{2r\tau}\\ && +C_r'C_2C_{\varepsilon}^{2r}\left(|u|^{2r(q-1)}+|u(.+he_j)|^{2r(q-1)}\right)\left(\left(e^{2r\gamma|{u}|^2}-1\right)\left(e^{2r\gamma u(.+he_j)|^2}-1\right)\right.\\ && \left.+\left(e^{2r\gamma|{u}|^2}-1\right)+\left(e^{2r\gamma u(.+he_j)|^2}-1\right)\right). \end{align}\] Thus, by H\(\ddot{\text{o}}\)lder’s inequality and Proposition 3 \[\begin{align} \int_{\mathbb{R}^2} |f(\tilde{u})|^{2r} & \leq & 2C_r\varepsilon^{2r}C_1 | u |_{2r\tau}^{2r\tau} +C_r'C_2C_{\varepsilon}^{2r}\int_{\mathbb{R}^2}|u|^{2r(q-1)}\left(e^{2r\gamma|{u}|^2}-1\right)\left(e^{2r\gamma u(.+he_j)|^2}-1\right)\\ &&+ C_r'C_2C_{\varepsilon}^{2r}\int_{\mathbb{R}^2}|u(.+he_j)|^{2r(q-1)}\left(e^{2r\gamma|{u}|^2}-1\right)\left(e^{2r\gamma u(.+he_j)|^2}-1\right)\\ && + C_r'C_2C_{\varepsilon}^{2r}\int_{\mathbb{R}^2}|u|^{2r(q-1)}\left(e^{2r\gamma|{u}|^2}-1\right)\\ && + C_r'C_2C_{\varepsilon}^{2r}\int_{\mathbb{R}^2}|u|^{2r(q-1)}\left(e^{2r\gamma|{u(.+he_j)}|^2}-1\right)\\ && + C_r'C_2C_{\varepsilon}^{2r}\int_{\mathbb{R}^2}|u(.+he_j)|^{2r(q-1)}\left(e^{2r\gamma|{u}|^2}-1\right)\\ && + C_r'C_2C_{\varepsilon}^{2r}\int_{\mathbb{R}^2}|u(.+he_j)|^{2r(q-1)}\left(e^{2r\gamma|{u(.+he_j)}|^2}-1\right)\\ & \leq & C_r'\varepsilon^{2r}C_1 |u|_{2r\tau}^{2r\tau} + C_3C_{\varepsilon}^2| u |_{4r(q-1)}^{2r(q-1)}\int_{\mathbb{R}^2}\left(e^{8r\gamma|u|^2}-1\right) \\ &&+ C_4C_{\varepsilon}^2 | u|_{4r(q-1)}^{2r(q-1)} \left(\int_{\mathbb{R}^2}\left(e^{4r\gamma|u|^2}-1\right)\right) <+\infty \text{ for all } |h|\leq h_0, \end{align}\] Hence \[\label{ebb} \sup_{|h|\leq h_0} \int_{\mathbb{R}^2}|f(\widetilde{u})|^{2r}\,dx<+\infty.\tag{34}\] Let \(\eta>0\). For any measurable set \(E\subset \mathbb{R}^2\), by Hölder’s inequality and 34 , \[\int_E |f(\widetilde{u})|^2\,dx \leq \left(\int_E |f(\widetilde{u})|^{2r}\,dx\right)^{1/r} |E|^{(r-1)/r} \leq C |E|^{(r-1)/r}.\] Hence, choosing \(\delta>0\) such that \(C\delta^{(r-1)/r}<\eta\), we obtain \[|E|<\delta \quad\Longrightarrow\quad \sup_{|h|\leq h_0}\int_E |f(\widetilde{u})|^2\,dx<\eta.\] Next, we need the tightness property, that is, for any \(R>0\), we should be able to find \(S_{R}\subset \mathbb{R}^2\) such that \(|S_{R}|<+\infty\) and \[\label{tightness} \sup \int_{\mathbb{R}^2\setminus S_R}|f(\tilde{u})|^2 \rightarrow 0 \text{ as } h\rightarrow 0.\tag{35}\] Now, for any ball \(B_{R}(0)\) of radius \(R\) centered at origin, if we denote \(A_R:=\mathbb{R}^2\setminus B_R(0)\), then we have \[\begin{align} \label{eq951} \int_{A_{R}} |f(\tilde{u})|^2 & \leq & 2 \varepsilon\int_{A_R}|\tilde{u}|^{2\tau} +4C_{\varepsilon} \int_{A_{R}}|\tilde{u}|^{2(q-1)}\left(e^{2\gamma|\tilde{u}|^2}-1\right)\nonumber\\ & \leq & 2\varepsilon\int_{A_{R}}|\tilde{u}|^{2\tau} +C_5 \left(\int_{A_R}|\tilde{u}|^{4(q-1)}\right)^{\frac{1}{2}}\left(\int_{A_{R}}\left(e^{4\gamma|\tilde{u}|^2}-1\right)\right)^{\frac{1}{2}}. \end{align}\tag{36}\] Further, since \(\tilde{u}\) lies in \(L^p(\mathbb{R}^2)\) for all \(p\geq 2\), by Dominated convergence theorem, \(|\tilde{u}|^p\chi_{\mathbb{R}^2\setminus B_R(0)}\rightarrow 0\) as \(R\rightarrow \infty\) in \(L^1(\mathbb{R}^2)\), that is \[\int_{A_R}|\tilde{u}|^p=\int_{\mathbb{R}^2}|\tilde{u}|^p\chi_{\mathbb{R}^2\setminus B_R(0)}dx \rightarrow 0 \text{ as } R\rightarrow \infty \text{ for all }p\geq 2 \text{ and } |h|\leq h_0,\] thus, \[\label{eq952} \sup_{|h|\leq h_0} \int_{A_R}|\tilde{u}|^p \rightarrow 0 \text{ as } R\rightarrow \infty \text{ for all } p\geq2;\tag{37}\] also, as done above in the uniform integrability case, we can find \(M>0\) such that \[\label{eq953} \int_{A_R}\left(e^{4\gamma|\tilde{u}|^2}-1\right)\leq M \text{ for all } h.\tag{38}\] Therefore, using 37 and 38 in 36 we get 35 with \(S_R=B_R(0)\). Hence, by Vitali’s convergence theorem, \(f(\tilde{u})\rightarrow f(u)\) in \(L^2(\mathbb{R}^2)\). ◻

Now, let us prove the Pohožaev identity ??

Proof of Theorem 6. The proof is directly followed from [14], thus, we omit the repetitive calculations and highlight the main difference that occurred due to the growth of the function \(f\). Considering the notations as in [14] we

estimate \(I_{3,2}+J_{2,2}\):
Now, using \(\displaystyle \phi_1=\sum_{i=1}^2 D_j(u)\) as test function, we get: \[\label{phi95195test} \int_{\mathbb{R}^2}\nabla u \nabla \phi_1 +\ll u, \phi_1\gg = \Lambda\int_{\mathbb{R}^2}(I_{\alpha}*F(u))f(u)\phi_1 -\lambda\int_{\mathbb{R}^2}u\phi_1.\tag{39}\] Then, as done in [38], since \[D_j(|\nabla u|^2)-2\nabla u \nabla D_j(\nabla u)\geq 0;\] \[D_j(|u(x)-u(y)|^2)-2(u(x)-u(y))(D_ju(x)-D_ju(y))\geq 0 \text{ for all } x, y \in \mathbb{R}^2,\] and \(x_j\psi_R(x)\leq \frac{2}{R}\) for all \(x\in \mathbb{R}^2\), we get: \[\begin{align} && |I_{3,2}+J_{2,2}| = \frac{1}{2} \left|\sum_{j=1}^2\int_{\mathbb{R}^2}\left(D_j(|\nabla u|^2)-2\nabla uD_j(\nabla u)\right)\psi_Rx_j dx \right.\\ && \;\;\left. +\sum_{j=1}^2\frac{1}{2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \frac{\left(D_j(|u(x)-u(y)|^2)-2(u(x)-u(y))(D_j u(x)-D_j u(y))\right)\psi_Rx_j}{|x-y|^{2+2s}}dxdy\right|\\ && \leq \frac{1}{R} \left|\sum_{j=1}^2\int_{\mathbb{R}^2}\left(D_j(|\nabla u|^2)-2\nabla uD_j(\nabla u)\right)dx \right.\\ && \;\;\left. +\sum_{j=1}^2\frac{1}{2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \frac{\left(D_j(|u(x)-u(y)|^2)-2(u(x)-u(y))(D_j u(x)-D_j u(y))\right)}{|x-y|^{2+2s}}dxdy\right|\\ & &= \frac{1}{R}\sum_{j=1}^2\left(\int_{\mathbb{R}^2}\left(D_j(|\nabla u|^2)-2\nabla uD_j(\nabla u)\right)dx\right.\\ && \;\;\left. +\frac{1}{2}\int_{\mathbb{R}^2}\int_{\mathbb{R}^2} \frac{\left(D_j(|u(x)-u(y)|^2)-2(u(x)-u(y))(D_j u(x)-D_j u(y))\right)}{|x-y|^{2+2s}}dxdy\right)\\ & &= -\frac{2}{R}\left(\int_{\mathbb{R}^2}\nabla u \sum_{j=1}^2D_j(\nabla u)+\sum_{i=1}^2 \ll u, D_j(u)\gg\right)\\ & &= -\frac{2}{R} \left(\int_{\mathbb{R}^2}\nabla u \nabla \phi_1+\ll u,\phi_1 \gg\right) = \frac{2}{R}\left(0-\int_{\mathbb{R}^2}(\Lambda(I_{\alpha}*F(u))f(u)-u)\phi_1\right). \end{align}\] Now, since \(\Lambda(I_{\alpha}*F(u))F(u)-u\in L^1(\mathbb{R}^2)\) we get: \[\int_{\mathbb{R}^2}D_j\left(\Lambda(I_{\alpha}*F(u))F(u)-\frac{|u|^2}{2}\right)dx=0,\] since for any integrable function \(g\), we have: \[\int_{\mathbb{R}^2}D_j(g)(x)dx= \frac{1}{h}\int_{\mathbb{R}^2}\left(g(x+he_j)-g(x)\right)dx = \frac{1}{h}\left( \int_{\mathbb{R}^2}g(x)dx-\int_{\mathbb{R}^2}g(x)dx\right)=0;\] and hence, \[\begin{align} \label{I95344295J952442} |I_{3,2}+J_{2,2}| & \leq & \frac{2}{R}\left(\int_{\mathbb{R}^2}\sum_{j=1}^2 D_j\left(\Lambda(I_{\alpha}*F(u))F(u)-\frac{|u|^2}{2}\right)dx-\int_{\mathbb{R}^2}(\Lambda(I_{\alpha}*F(u))f(u)-u)\phi_1\right)\nonumber\\ & = & \frac{2}{R} \sum_{j=1}^2 \int_{\mathbb{R}^2}\left(D_j\left(\Lambda(I_{\alpha}*F(u))F(u)-\frac{|u|^2}{2}\right)-(\Lambda(I_{\alpha}*F(u))f(u)-u)D_j(u)\right)dx\nonumber\\ & = & \frac{2}{R}\left(\sum_{j=1}^2 A^1_j - A^2_j\right), \end{align}\tag{40}\] where \[A_j^1= \Lambda\int_{\mathbb{R}^2}\left(D_j((I_{\alpha}*F(u))F(u)-(I_{\alpha}*F(u))f(u)D_j(u)\right)dx;\] and \[A_j^2=\int_{\mathbb{R}^2}\left(D_j\left(\frac{|u|^2}{2}\right)-uD_j(u)\right)dx.\] Now, since \((I_{\alpha}*F(u))\in L^{\infty}(\mathbb{R}^N)\) and \(F\) is convex, by using mean value theorem, we get \[\begin{align} A_j^1 & = & \Lambda \int_{\mathbb{R}^2}\left(D_j((I_{\alpha}*F(u))F(u)-(I_{\alpha}*F(u))f(u)D_j(u)\right)dx\\ & \leq & C\Lambda\int_{\mathbb{R}^2}\left(\frac{F(u)(x+he_j)-F(u)(x)}{h}-f(u)D_j(u)\right)dx\\ & = & C\Lambda\int_{\mathbb{R}^2}\left(D_j(F(u))-f(u)D_j(u)\right)dx\\ & = & C\Lambda\int_{\mathbb{R}^2}\left(f(\tilde{u})D_j(u)-f(u)D_j(u)\right)dx, \end{align}\] where \(\tilde{u}\) lies between \(u\) and \(u(.+he_j)\) Thus \[\label{A95j951} A_j^1 \le C\Lambda\int_{\mathbb{R}^2}(f(\tilde{u})-f(u))D_j(u)dx;\tag{41}\] also, by the mean value theorem: \[\label{A95j952} A_j^2 = \int_{\mathbb{R}^2}(\tilde{u}-u)D_j(u).\tag{42}\] Therefore, using 41 and 42 in 40 , we get: \[|I_{3,2}+J_{2,2}| \leq C_1 \left(\sum_{j=1}^2\Lambda\int_{\mathbb{R}^2}(f(\tilde{u})-f(u))D_j(u)dx + \sum_{j=1}^2\int_{\mathbb{R}^2}(\tilde{u}-u)D_j(u)dx\right).\] Here, since \(D_j u \rightarrow \frac{\partial u}{\partial x_j}\) and \(\tilde{u}\rightarrow u\) as \(h\rightarrow 0\), in \(L^2(\mathbb{R}^N)\) we get: \[\sum_{j=1}^2\int_{\mathbb{R}^2}(\tilde{u}-u)D_j(u)dx \le \sum_{j=1}^{2}\left(\int|\tilde{u}-u|^2\right)^{\frac{1}{2}}\left(\int |D_ju|^2\right)^{\frac{1}{2}} \rightarrow 0 \text{ as } h\rightarrow 0.\] Next, by Proposition 8 we have \[\begin{align} \sum_{j=1}^2\int_{\mathbb{R}^2}(f(\tilde{u})-f(u))D_j(u)dx & \le & \sum_{j=1}^2 \left(\int_{\mathbb{R}^2}|f(\tilde{u})-f(u)|^2\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^2}|D_ju|^2\right)^{\frac{1}{2}}\rightarrow 0, \end{align}\] as \(h\rightarrow0\). Thus, \(|I_{3,2}+I_{2,2}|\rightarrow 0\). Further, following the proof of [14] step by step, we get ?? . ◻

Declarations↩︎

Ethical Approval. Not applicable.

Competing interests. The authors declare that they have no competing interests.

Authors contributions. The authors contributed equally to this work.

Availability of data and materials. Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Acknowledgements↩︎

The author, Nidhi Nidhi (PMRF ID-142685), acknowledges the support provided by the Ministry of Education, Government of India, under the Prime Minister’s Research Fellows scheme.

The author, Lovelesh Sharma, gratefully acknowledges the support received under the project at IIT Delhi (Project No. PD16047).

The author K. Sreenadh thanks the Department of Science and Technology (DST) India, for providing support under the Improvement of S&T Infrastructure (FIST) programme. (Project No. SR/FST/MS-1/2019/45).

References↩︎

[1]

S. Deng, J. Yu, Normalized solutions for a Choquard equation with exponential growth in \(\mathbb{R}^{2}\), ZAMP 74(3) (2023), Article 103.

[2]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20(1971), 1077–1092.

[3]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17(1967), 473–483.

[4]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57(2) (1977), 93–105.

[5]

R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativity Gravitation 28(1996), 581–600.

[6]

R. Filippucci, M. Ghergu, Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms, Nonlinear Anal. 197(2020), 111857.

[7]

V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265(2) (2013), 153–184.

[8]

Z. Liu, V. D. Rădulescu, C. Tang, J. Zhang, Another look at planar Schrödinger–Newton systems, J. Differential Equations 328(2022), 65–104.

[9]

S. Dipierro, E. P. Lippi, E. Valdinoci, (Non)local logistic equations with Neumann conditions, Ann. Inst. H. Poincaré C 40(5) (2022), 1093–1166.

[10]

S. Dipierro, E. Valdinoci, Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes, Physica A 575(2021), 126052.

[11]

B. Pellacci, G. Verzini, Best dispersal strategies in spatially heterogeneous environments, J. Math. Biol. 76(6) (2018), 1357–1386.

[12]

G. Pagnini, S. Vitali, Should I stay or should I go? Zero-size jumps in random walks for Lévy flights, Fract. Calc. Appl. Anal. 24(1) (2021), 137–167.

[13]

S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, A Brezis–Nirenberg type result for mixed local and nonlocal operators, NoDEA Nonlinear Differential Equations Appl. 32(4) (2025), Article 62.

[14]

G. C. Anthal, P. Garain, Nidhi, Ground state solutions of mixed local-nonlocal equations with Hartree type nonlinearities, arXiv:2602.02168 (2026).

[15]

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28(10) (1997), 1633–1659.

[16]

T. Bartsch, S. de Valeriola, Normalized solutions of nonlinear Schrödinger equations, Arch. Math. (Basel) 100(1) (2013), 75–83.

[17]

B. Noris, H. Tavares, G. Verzini, Existence and orbital stability of the ground states with prescribed mass, Anal. PDE 7(8) (2015), 1807–1838.

[18]

D. Pierotti, G. Verzini, Normalized bound states for the nonlinear Schrödinger equation, Calc. Var. Partial Differential Equations 56(2017), Article 133.

[19]

T. Gou, L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity 31(5) (2018), 2319.

[20]

T. Bartsch, L. Jeanjean, N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on \(\mathbb{R}^{3}\), J. Math. Pures Appl. 106(4) (2016), 583–614.

[21]

T. Bartsch, L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems, Proc. Roy. Soc. Edinburgh Sect. A 148(2) (2018), 225–242.

[22]

B. Noris, H. Tavares, G. Verzini, Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity 32(3) (2019), 1044.

[23]

J. Giacomoni, Nidhi, K. Sreenadh, Normalized solutions to a Choquard equation involving mixed local and nonlocal operators, NoDEA Nonlinear Differential Equations Appl. 32(6) (2025), Article 127.

[24]

J. Giacomoni, Nidhi, K. Sreenadh, Normalized solutions to a critical growth Choquard equation involving mixed operators, Asymptot. Anal. 143(3) (2025), 871–899.

[25]

B. Pellacci, A. Pistoia, G. Vaira, G. Verzini, Normalized concentrating solutions to nonlinear elliptic problems, J. Differential Equations 275(2021), 882–919.

[26]

S. Deng, J. Yu, Normalized solutions for a Choquard equation with exponential growth in \(\mathbb{R}^{2}\), Z. Angew. Math. Phys. 74(2023).

[27]

L. Shen, M. Squassina, Concentrating normalized solutions for 2D nonlocal Schrödinger equations, Electron. J. Differential Equations 2025(2025), Article 34.

[28]

R. Yi, A. Qian, Normalized solutions to a class of Kirchhoff equation with general nonlinearity, Bound. Value Probl. (2026).

[29]

P. Jin, X. Tang, D. Hu, Solutions with prescribed mass for nonautonomous critical Kirchhoff equations with convolution nonlinearity, Discrete Contin. Dyn. Syst. B 31(2026), 277–303.

[30]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69(1) (1979), 19–30.

[31]

G. Leoni, A first course in fractional Sobolev spaces, American Mathematical Society, vol 229, 2023.

[32]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb{R}^2\), Comm. Partial Differential Equations 17(3–4) (1992), 407–435.

[33]

D. Cassani, F. Sani, C. Tarsi, Equivalent Moser type inequalities in \(\mathbb{R}^2\) and the zero mass case, J. Funct. Anal. 267(11) (2014), 4236–4263.

[34]

E. H. Lieb, M. Loss, Analysis, AMS, Providence, RI, 2001.

[35]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, AMS, 2003.

[36]

O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer, 1993.

[37]

C. O. Alves, D. Cassani, C. Tarsi, M. B. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \(\mathbb{R}^{2}\), J. Differential Equations 261(3) (2016), 1933–1972.

[38]

G. C. Anthal, P. Garain, Pohožaev-type identities for classes of quasilinear elliptic local and nonlocal equations and systems, with applications, arXiv:2506.08667 (2025).

[39]

S. Kesavan, Topics in Functional Analysis and Applications, New Age International, New Delhi, 2019.

[40]

M. G. Garroni, J. L. Menaldi, Second order elliptic integro-differential problems, Chapman & Hall/CRC, 2002.

Normalized solutions to an exponential growth Choquard equation driven by mixed local-nonlocal operator in \(\mathbb{R}^2\)